Embed Size (px)
Citation preview
ABSTRACT
Title of dissertation: TEACHING FOR INCLUSION: THE EFFECTS OF A PROFESSIONAL DEVELOPMENT COURSE FOR SECONDARY GENERAL AND SPECIAL EDUCATION MATHEMATICS TEACHERS FOR INCREASING TEACHER KNOWLEDGE AND SELF-EFFICACY IN GEOMETRY
Kenneth Wright, Doctor of Philosophy, 2015 Dissertation directed by: Professor Peter Leone
Department of Counseling, Higher Education, and Special Education
The current study examined the effects of a co-teacher professional development
course for increasing the knowledge and self-efficacy of special education and general
education geometry teachers in an inclusion setting. The professional development course
included instruction on Universal Design for Learning instructional strategies as well as
similarity and congruence in geometry. The course was presented in a blended learning
format and included in-person and online activities. A multiple probe design across three
sets of two teachers for a total of 6 participants was used in this study to demonstrate a
functional relationship between the independent and dependent variables. The
participants were six special education and general education geometry teachers from
public charter schools in Washington, DC. Results of the study demonstrated that
participants were able to improve their content and pedagogical content knowledge in
geometry as well as their self-efficacy for teaching in an inclusion setting. Specifically,
special education teachers demonstrated a greater increase in content knowledge while
general education teachers demonstrated a greater increase in self-efficacy for teaching
students with disabilities. The study suggests that providing professional development for
co-teachers can enhance collaboration as well as increase content knowledge and teacher
self-efficacy.
TEACHING FOR INCLUSION: THE EFFECTS OF A PROFESSIONAL DEVELOPMENT COURSE FOR SECONDARY GENERAL AND SPECIAL EDUCATION MATHEMATICS TEACHERS FOR
INCREASING TEACHER KNOWLEDGE AND SELF-EFFICACY IN GEOMETRY
by
Kenneth Wright
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment of the requirements for the degree of
Doctor of Philosophy 2015
Advisory Committee:
Professor Peter E. Leone, Chair
Professor Lawrence Clark
Professor Andrew Egel
Professor Susan De La Paz
Professor Whitney Johnson
ii
Acknowledgements
I want to thank my Committee Chair Dr. Peter Leone for coming in and helping me get across
the finish line. Special thanks to Dr. Paula Maccini for her time, guidance, and support from the
very beginning. I would also like to express sincere appreciation to Dr. Lawrence Clark for his
guidance and encouragement, Dr. Whitney Johnson for teaching me creative math instructional
strategies over the summer, and Dr. Andy Egel for his feedback on my design approach.
To the students, the special education department, and the charter school leaders where I
conducted my study, you have played a huge role in my career development and the success of
this study. My final thanks go to my sons Kenny and Darren, and to the love of my life, my
fiancée` Nicole Bryan for her love, support and encouragement.
iii
Table of Contents
Acknowledgements ............................................................................................................... ii
Table of Contents ................................................................................................................. iii
List of Tables ......................................................................................................................... v
List of Figures ....................................................................................................................... vi
Chapter 1: Introduction ......................................................................................................... 1 Status of Mathematics Proficiency in the U.S. .............................................................................................................. 3 Characteristics of Students with LD ................................................................................................................................. 6 Reform Movement................................................................................................................................................................... 7 Inclusion, Teacher Knowledge, and Self-Efficacy ..................................................................................................... 12 Teacher Professional Development ............................................................................................................................... 15 Statement of Purpose ........................................................................................................................................................... 17 Research Questions and Hypotheses ............................................................................................................................ 18 Definition of Key Terms ...................................................................................................................................................... 20
Chapter 2: Review of the Literature ..................................................................................... 22 Proficiency of U.S. Students in Mathematics .............................................................................................................. 23 Characteristics of Students with Learning Disabilities .......................................................................................... 24 Universal Design for Learning .......................................................................................................................................... 32 Teacher Self-Efficacy ............................................................................................................................................................ 34 Content and Pedagogical Content Knowledge ........................................................................................................... 37 Organization of the Review of the Literature on Professional Development ............................................... 39 Overview of the Studies ...................................................................................................................................................... 40 Content Analysis .................................................................................................................................................................... 41 Professional Development for Special Education Teachers ................................................................................ 50 Video Cases .............................................................................................................................................................................. 51 Animated Representations of Teaching ....................................................................................................................... 52 Study Implications ................................................................................................................................................................ 54
Chapter 3: Methodology ...................................................................................................... 56 Professional Development Resources........................................................................................................................... 58 Experimental Design and Study Procedures .............................................................................................................. 60 Pre- and Post-Study Instruments .................................................................................................................................... 61 Single Subject Data Collection .......................................................................................................................................... 64 Data analysis procedures ................................................................................................................................................... 69
Chapter 4: Results ............................................................................................................... 72 Participant Profile ................................................................................................................................................................. 72 Results on Professional Development Course ........................................................................................................... 73 Research Question 1 ............................................................................................................................................................. 73 Pre-/Post- Survey .................................................................................................................................................................. 73 Domain and Objective Probes .......................................................................................................................................... 74 Research Question 2 ............................................................................................................................................................. 78 Research Question 3 ............................................................................................................................................................. 79 Social Validity .......................................................................................................................................................................... 80
Chapter 5: Discussion .......................................................................................................... 82
iv
Study Design and Rationale ............................................................................................................................................... 82 Interpretation of Findings.................................................................................................................................................. 84 Summary ................................................................................................................................................................................... 87 Study Limitations................................................................................................................................................................... 87 Directions for Future Research ........................................................................................................................................ 88 Conclusion ................................................................................................................................................................................ 90
Appendix A: Principal Letter ................................................................................................ 95
Appendix B: Participant Consent Form ................................................................................. 96
Appendix C: UDL Guidelines ................................................................................................. 98
Appendix D: Table of Measures ........................................................................................... 99
Appendix E: Teacher Sense of Efficacy Scale – Short Form and Permission Letter ................ 100
Appendix F: Sample GAST Items ........................................................................................ 102
Appendix G: Focus Group Questions .................................................................................. 103
Appendix H: Sample Domain Probe................................................................................... 104
Appendix I: Sample Objective Probe ................................................................................. 106
Appendix J: Course Outline ................................................................................................ 107
Appendix K: Sample Lesson Plan ........................................................................................ 109
Appendix L: Fidelity of Treatment Checklist ....................................................................... 113
Appendix M: Social Validity Measure ................................................................................. 115
References ........................................................................................................................ 117
v
List of Tables
1. Summary of Studies 104
2. Participant Demographic Information 56
3. Data Sources and Analyses 69
4. TSES Results 73
5. Within Group Pre and Post Mean Scores on GAST Assessment of Congruence
and Similarity
77
6. Average Pre and Post TSES Scores By Dimension 77
7. Social Validity Measure Results 79
vi
List of Figures
1. Animated Script Example 69 2. Percentage of Accurate Responses on Domain Probes 74 3. Percentage of Accurate Responses on Objective Probes 76
1
Chapter 1: Introduction
Mathematical literacy has critical implications for personal success in post-
secondary education, career opportunities, and the ability to increase future income. The
Final Report of the National Mathematics Advisory Panel (NMAP, 2008) reported a high
correlation between completion of an Algebra II course with success in post secondary
education and earnings from employment. Specifically, completion of Algebra II more than
doubles students’ chances of graduating from college as compared to students with less
mathematical preparation. Mathematics achievement is also regarded as an indicator of the
nation’s ability to compete in a global economy. Global leadership is buttressed by an
educated technical workforce in today’s era of technological advancements (NMAP, 2008).
Geometry, in particular, is considered to be a meaningful and vital topic in
mathematics for it addresses concepts that are useful for many careers, daily living skills,
and provides a critical foundation for the study of algebra (Cawley, Foley, & Hayes, 2009;
NMAP, 2008). By engaging with geometry content, students have a number of
opportunities to develop cognitive, language, and communication processes that develop
critical thinking and problem solving skills (Seago, Driscoll, & Jacobs, 2010). In addition,
geometry involves measurement and assigning characteristics to objects, skills used daily
in living and employment. Congruence and similarity, in particular, are considered
important topics within geometry (CCSSI, 2010; Seago, Driscoll, & Jacobs, 2010).
Understanding congruence and similarity is useful for a variety of real life applications
including following directions for assembling furniture, understanding the blueprint of a
building, or reading a scale on a map. Given the importance of mathematical competency, it
is imperative that all students have access to quality mathematics instruction.
2
During the past 10 years, the percentage of students in special education in inclusive
classrooms has increased considerably, with over half of all identified students being
educated in general education settings (National Center for Educational Statistics, 2011).
Nearly 61% of students diagnosed with specific learning disability (SLD) spend 80% or
more of each school day in a general education classroom and are not exempt from state
and national standardized assessments (U.S. Department of Education, 2011). One of the
premises of inclusion is that students with disabilities receive access to the same
curriculum as their non-disabled peers, the goal being to improve the social and academic
outcomes for students who with disabilities. However, students with a learning disability
(LD) often experience academic failure in general education classrooms (Schumaker &
Deshler, 1988). Abstract mathematical concepts, such as congruence and similarity, in
particular, pose a challenge for students with LD who possess a variety of characteristics
that make learning and remembering mathematics difficult (Miller & Mercer, 1997). Given
that recent legislation, such as the No Child Left Behind Act (NCLB) of 2001, mandates that
all students, including students with disabilities, demonstrate proficiency with the general
education curriculum, it is critical that close attention is given to practices and methods for
ensuring that all students achieve an understanding of content.
One of the implications for increasing the successful inclusion of students with
disabilities in general education classes is that teachers must adapt their instruction to
meet the needs of a more diverse group of students. Secondary content-area teachers, in
particular, often feel ill prepared to teach students with disabilities or know how to provide
the necessary accommodations for students to access the curriculum due to little or no
training (Kosko & Wilkins, 2009; Maccini & Gagnon, 2006). In addition, special education
3
teachers, who are often hired to teach alongside general educators, generally find that they
lack the content knowledge for helping students meet the rigorous standards for
demonstrating proficiency (Eisenman, Pleet, Wandry, & McGinley, 2010). Thus, teachers
need intensive and sustained professional development for adapting instruction to meet
the needs of students with disabilities as well as increase their knowledge for teaching
mathematical concepts.
In the section below, I will first summarize the results from international, national,
and state mathematics assessments, particularly as they relate to geometry, followed by an
overview of mathematics reform efforts related to teaching and learning for all students. I
then discuss the characteristics of students with LD with an emphasis on the implications
for educating these students with their non-disabled peers. I will also discuss the role of
teacher knowledge as it relates to teacher efficacy and how in-service training may
increase the teacher’s ability to meet the needs of a diverse learning community. The
chapter concludes with a purpose statement, guiding research questions, and definitions of
key terminology.
Status of Mathematics Proficiency in the U.S.
Although researchers, policymakers, and other stakeholders have acknowledged the
importance of mathematical competency, students in the U.S. are not performing well as
reported on international, national, and state assessments. In terms of international
assessments, the Program for International Student Assessment (PISA) is sponsored by the
Organization for Economic Cooperation and Development (OECD) and measures the
mathematics literacy of 15-year-olds. The most recent PISA (2012) reported that U.S. 15-
year-olds scored significantly below the OECD average as they were outperformed by 25 of
4
the 34 countries (OECD, 2013). Further, only 12.6% of U.S. students scored at or above
Level 5 proficiency level, which represents the level at which students are able to select,
compare, and evaluate appropriate problem-solving strategies for dealing with complex
problems, skills essential in the study of geometry. A second international assessment, the
Trends in International Mathematics and Science Study (TIMSS), measures the
mathematics performance of students based on classroom curricula. Unlike the PISA
assessment, which assesses students’ ability to apply mathematical reasoning and problem
solving skills to real-life contexts, the TIMSS focuses on students’ knowledge of
mathematical concepts taught in the classroom. In 2011, students’ overall mathematics
performance at the eighth grade level was reported for 48 countries. Overall, the average
scale score for U.S. eighth-grade students was nine percentage points higher than the
international average and lagged behind 8 countries including Korea, Singapore, and
Chinese Taipei (Mullis, Martin, Foy & Arora, 2012). In addition, only 6% of U.S. eighth-
graders scored at or above the TIMSS advanced benchmark, which is the level at which
students can make generalizations, organize information, and solve complex problems. In
regard to the geometry domain, U.S. eighth-grade students scored 20 percentage points
below the average range, outperforming 30 countries, and scoring lower than 17 countries
(Mullis et al, 2012). The TIMSS geometry content covers congruence, similarity, and
application of geometric properties. Data for students with disabilities was not
disaggregated; therefore, no comparisons can be made between students with disabilities
and their international counterparts.
In addition to the overall underperformance of U.S. students in mathematics on
international assessments, U.S. students have not made the achievement gains expected on
5
national assessments, such as the National Assessment of Educational Progress (NAEP),
which covers several mathematics content areas, including geometry. Overall,
approximately two-thirds (64%) of eighth graders in the general education population
scored below the proficient level (i.e., competency in grade level material) on the most
recent NAEP assessments (National Center for Education Statistics, 2013). On the geometry
portion of the same test, the average scale score was 17 percentage points below the
proficient level for eighth-grade students (National Center for Education Statistics, 2013).
The performance results for secondary students with disabilities are more alarming.
On the same assessment, 93% of eighth-graders with disabilities performed below the
proficient level (National Center for Education Statistics, 2013). Further, the average scale
score for eighth grade students with disabilities on the geometry portion of the assessment
was 31 points lower than their general education peers (National Center for Education
Statistics, 2013). Students have also not performed well on state and district assessments.
In the District of Columbia, for example, 49% of elementary students and 46% of secondary
students achieved proficiency or above on the 2013 District of Columbia Comprehensive
Assessment, failing to meet the targets for Adequate Yearly Progress (Office of the State
Superintendent, n.d.). For students with disabilities, only 24% of elementary students and
16% of secondary students achieved proficiency. In Maryland, while 67% of all eighth
grade students demonstrated proficiency on the 2013 Maryland State Assessment, only
25% of eighth grade students with disabilities achieved the proficient level on the
assessment (Maryland Department of Education, 2013). Although there may be several
reasons to explain the low proficiency rates of students with disabilities in mathematics,
6
particular challenges associated with learning disabilities pose difficulties for children in an
educational setting.
Characteristics of Students with LD
Students with LD possess several characteristics that hinder the learning process and
impede their performance in mathematics. A learning disability is defined as a disorder in
one or more of the basic psychological functions involved in understanding or in using
language and manifests itself in an imperfect ability to listen, speak, read, write, spell, or do
mathematical calculations (IDEA, 2004). Students with LD in mathematics often struggle
with basic mathematics content due to poor short-term memory processes (Calhoon,
Emerson, Flores, & Houchins, 2007). For example, students with LD may have trouble
remembering the procedures for using the order of operations. Moreover, they may exhibit
reading and language difficulties. Specifically, students with LD typically have reading
problems, which may interfere with understanding mathematics vocabulary as well as
completing homework and textbook assignments (Steele, 2010). In geometry, language
deficits may also interfere with a student’s ability to associate words to the symbols of
geometric terms and also to respond to questions orally in class such as explaining the
difference between complementary and supplementary angles or discussing the use of the
Pythagorean Theorem (Bley & Thornton, 2001; Ives, 2007; Steele, 2010). In addition,
students with LD exhibit poor metacognition (Bley & Thornton, 2001). Poor metacognition
impedes students’ ability to engage in problem solving as well as to solve complex, multi-
step problems (Vaidya, 1999). For example, students with LD may have trouble
determining the value of the variable in an unknown angle measurement of a given
geometric figure.
7
Visual processing problems such as difficulty with visual spatial processing are also
common to many students with LD (Lerner & Johns, 2009) and can interfere significantly
with mathematics performance. For example, a student with visual spatial deficits may
have problems with identifying or demonstrating translations, rotations, and reflections.
Memory problems, typical for students with LD, can also make high school mathematics
courses challenging (Lerner & Johns, 2009). These difficulties often pose a challenge for
content-area teachers who are inadequately trained to provide instruction to students with
disabilities.
Reform Movement
To address the underperformance of U.S. students in mathematics, policymakers
have passed legislation and issued directives in an effort to improve mathematics
education. The National Council of Teachers of Mathematics (NCTM) continues to
spearhead efforts and guidance on mathematics content, pedagogy, and assessment across
the K-12 grade levels. The Principles and Standards for School Mathematics (NCTM, 2000)
maintains the principle of equity, demanding high expectations for mathematics learning
and access to high quality instruction and resources for all students. In addition, the
Professional Standards for Teaching Mathematics (NCTM, 1991) emphasizes the role of
teachers in the development of positive attitudes towards mathematics. According to this
document, it is essential that teachers provide a learning environment that facilitates and
promotes students' flexibility in thinking, creativity, perseverance, and self-confidence in
doing mathematics.
At the federal level, the reauthorization of the Elementary and Secondary Education
Act (2001), also known as No Child Left Behind (NCLB), contains several mandates for
8
ensuring quality instruction for all students, including students with disabilities. NCLB
requires that students receive instruction from a “highly qualified teacher,” defined as one
who holds a "bachelor's degree, has full state certification and has demonstrated subject
area competence for each subject taught" (NCLB, 2002). This mandate supports the idea
that teacher content knowledge plays a critical role in improving student achievement
(Darling-Hammond & Bransford, 2005; Hill, Rowan, & Ball, 2005). NCLB also includes
regulations that require states to submit plans for ensuring that students with disabilities
have access to the general education curriculum (NCLB, 2002).
Most recently, the National Governors Association Center for Best Practices (NGA
Center) and the Council of Chief State School Officers (CCSSO) have led the initiative to
develop a common set of English language arts and mathematics standards to be adopted
by all states. Known as the Common Core State Standards Initiative (CCSSI), the standards
address what students are expected to learn in order to prepare them for college and
career (CCSSI, 2010). To date, forty-five states have adopted this initiative. The goal of the
CCSSI for mathematics is to provide more clarity and coherence of what students should
know and understand in mathematics. The standards for geometry, in particular, cover
several critical areas including congruence and similarity, properties of two-dimensional
and three-dimensional shapes, applying the Pythagorean theorem, and application of
probability concepts (CCSSI, 2010).
Under the CCSSI are specific habits and practices that students should demonstrate
as they learn mathematical concepts across grade levels. The Standards for Mathematical
Practice describe varieties of expertise that mathematics educators at all levels should seek
to develop in their students. These practices rest on important “processes and
9
proficiencies” with longstanding importance in mathematics education. The first of these
are the NCTM process standards of problem solving, reasoning and proof, communication,
representation, and connections. The second are the strands of mathematical proficiency
specified in the National Research Council’s report Adding It Up: adaptive reasoning,
strategic competence, conceptual understanding (i.e., comprehension of mathematical
concepts, operations and relations), procedural fluency (i.e., skill in carrying out
procedures flexibly, accurately, efficiently and appropriately), and productive disposition
(i.e., habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled
with a belief in diligence and one’s own efficacy). The eight Common Core Standards for
Mathematical Practice include the following:
Make sense of problems and persevere in solving them. This practice standard
involves students performing actions such as explaining the meaning of a problem to
themselves, looking for entry points, and making conjectures and a plan to solve a problem.
(CCSS, 2010).
Reason abstractly and quantitatively. This practice standard involves students
demonstrating the ability to create coherent representations of problems, attend to the
meaning of quantities, use properties of operations and objects, decontextualize situations
by manipulating mathematical symbols abstractly without necessarily attending to the
original referents, and to pause and contextualize the symbols during the manipulation to
make sense of the symbols concretely in terms of the original context (CCSS, 2010).
Construct viable arguments and critique the reasoning of others. Students who
demonstrate this standard are able to understand stated assumptions/definitions, make
conjectures, build progressions of statements to explore the truth of conjectures, justify
10
conclusions, compare the effectiveness of two plausible arguments, and distinguish and use
correct logic and reasoning. (CCSS, 2010).
Model with mathematics. This practice standard involves students applying
mathematics to solve everyday problems, using multiple representations such as graphs,
tables and equations and interpreting/reflecting on the mathematical results in the context
of the situation (CCSS, 2010).
Use appropriate tools strategically. This practice standard involves students
considering and using available tools (e.g., concrete model, ruler, compass, calculator,
spreadsheets, dynamic geometry software) and evaluating their effectiveness for a given
situation to help students explore and deepen their understanding of mathematical
concepts (CCSS, 2010).
Attend to precision. This practice standard involves students using clear and
precise definitions in discussions and explanations of their own work, stating the meaning
of and/or correctly using symbols specifying units of measure, calculating with precision
that is appropriate for a problem context, and giving carefully formulated explanations to
each other (CCSS, 2010).
Look for and make use of structure. This practice standard involves students
analyzing objects and examples to discern patterns and structures, extending lines in
geometric figures to assist in problem solving, and “seeing” complicated things such as an
algebraic expression as either a single whole or composed of several objects (CCSS, 2010).
Look for and express regularity in repeated reasoning. This practice standard
involves students noticing repeated calculations and looking for general methods or
11
shortcuts, maintaining oversight of the problem solving process, and continually evaluating
reasonableness of approach and results (CCSS, 2010).
As addressed in the CCSSI, students eligible for special education services under the
Individuals with Disabilities Education Act (IDEA) must be able to access the general
education curriculum and be prepared for post-secondary success in college and career.
Although the CCSSI does not prescribe specific strategies or methods of instruction, the
Standards do include guidance on accommodating students with disabilities and several
organizations, including the Council for Exceptional Children, have voiced their support for
this effort (CCSSI, 2010). Given the wide acceptance of these standards, it is imperative that
teachers receive professional development for increasing their knowledge of geometry
content as many states follow the traditional secondary mathematics curriculum sequence
that includes Algebra I, Geometry, Algebra II, and Pre-Calculus (CCSSI, 2010). There are
also implications for teachers to adjust their instruction to help students meet the new
curriculum demands under the Common Core Standards (CCSSI, 2010).
The CCSSI reference Universal Design for Learning (UDL) as guiding principles for
teachers to use for ensuring all students have access to the common core learning
standards. UDL expands upon the construct of Universal Design in architecture and
product design, and was established by Rose and Meyer (2002) following the 1997
reauthorization of the Individuals with Disabilities Education Act ([IDEA], Edyburn, 2010).
UDL is a set of principles for curriculum development that give all individuals equal
opportunities to learn. It provides a blueprint for creating instructional goals, methods,
materials, and assessments that work for everyone--not a single, one-size-fits-all solution
but rather flexible approaches that can be customized and adjusted for individual needs.
12
Thus, teachers will find these principles helpful when planning their instruction for
students with diverse learning needs.
Inclusion, Teacher Knowledge, and Self-Efficacy
Due to recent policy reforms and legislative mandates, schools are increasingly held
accountable for demonstrating a positive impact on achievement for all students (Darling-
Hammond, 2004). In response, schools are increasingly educating students with disabilities
alongside their non-disabled peers in an inclusive classroom setting, presenting several
challenges for general education and special education teachers alike. Traditionally,
personnel from these two educational fields receive separate training as general education
teachers are prepared through a focus on content and pedagogy related to their expertise
(e.g., mathematics) while special educators are trained on best practices for delivering
specialized instruction across multiple subject areas (Buell, Hallam, Gamel-McCormick, &
Scheer, 1999). However, with the increase in inclusionary practices, it is essential to
address the needs and supports of both general and special educators who are often
required to work in tandem to meet the diverse learning needs of the students they teach.
Ensuring that students with and without disabilities receive high-quality instruction often
relies on the successful preparation and support of personnel from both systems (Villa,
1996).
Teachers’ perception of their self-efficacy plays a critical role in the instructional
decisions they make (Kosko & Wilkins, 2009). Researchers (Bandura, 1994; Hoy, 2007)
describe self-efficacy as one’s perceived level of competency and ability to carry out
specific behaviors that influence desired outcomes for student learning and note that
teacher efficacy is an important variable in teacher effectiveness. Teachers must have
13
confidence in their ability to influence the achievement of their students. The implications
for possessing high levels of efficacy are evident in research (Gibson & Dembo, 1984;
Swackhamer, Koellner, Basile, Kimbrough, 2009). Specifically, it was reported that teachers
with high self-efficacy were more open to using reform-based instructional methods
(Czerniak & Chiarleott, 1990), more apt to work with students who have difficulties, and
more open to adapting their instruction to better support their students (Gibson & Dembo,
1984; Swackhamer, Koellner, Basile, Kimbrough, 2009). However, much of the research on
teacher’s self-efficacy involves pre-service teachers (Kosko & Wilkins, 2009). Future
research is needed to fill the gap in the literature by focusing on inservice special education
and general education mathematics teachers.
In order to improve mathematics instruction and increase the use of effective
teaching practices, teachers must possess extensive knowledge of mathematics content as
well as an understanding of how students think and learn (Feuerborn, Chinn, & Morlan,
2009; Heck, Banilower, Weiss, & Rosenberg, 2008). Shulman (1986) proposed that such
teacher knowledge is represented in three categories: (a) general pedagogical knowledge,
which involves general classroom management strategies and organization that transcend
subject matter; (b) content knowledge, which includes the knowledge and understanding
of mathematical facts, skills, and concepts that students need to learn; and (c) pedagogical
content knowledge which involves the ability to present mathematical knowledge in an
effective way for students to learn. Recent mathematics research (Krauss, et al., 2008;
Shulman, 1986) has generally focused on the development of teachers’ content knowledge
(CK). Teachers with a stronger conceptual understanding of mathematics content are
better equipped to provide higher quality instruction when compared with teachers with
14
only a limited understanding of the material (Ball & Cohen, 1999; Garet et al., 2001; Krauss
et al., 2008).
Research demonstrates a positive correlation between teacher CK and student
achievement (Hill, Rowan, & Ball, 2005; Mullens & Murnane, 1996). For example, in a four-
year study of 115 elementary schools, Hill, Rowan, and Ball (2005) found that teacher’s
mathematical knowledge was significantly related to student achievement at first and third
grades. Researchers defined teacher’s mathematical knowledge as the teacher’s ability to
solve, explain, and interpret mathematical concepts, which goes beyond the teacher’s
computational skills or years of coursework. In a one-year study of tenth grade students
and their mathematics teachers in Germany, Bruner and colleagues (2009) investigated the
importance of CK and pedagogical content knowledge (PCK) for improving student
outcomes in mathematics based on a test covering the federal states curriculum standards
for tenth grade. Although researchers determined that PCK had a greater effect on student
achievement, CK was highly correlated with PCK and it was concluded that CK helps to
form the basis for the development of PCK.
Given the importance of increasing teacher knowledge of mathematics content and
pedagogy for improving student outcomes, researchers and policy makers have sought to
improve the quality and number of mathematics courses required in pre-service teacher
education programs (National Commission on Teaching and America’s Future, 1996).
However, there remains a need for increased training opportunities for practicing
mathematics teachers (Feuerborn, et al., 2009; Kosko & Wilkins, 2009). Inservice teachers
need meaningful professional development that expands their knowledge and
15
understanding of mathematics content and pedagogy for increasing their ability to meet
the needs of all learners.
Teacher Professional Development
A growing body of research demonstrates that teacher professional development
(PD) can be an effective means for improving teacher CK and PCK in mathematics (Garret,
Porter, Desimone, Birman & Kwang Suk, 2001; Heck, Banilower, Weiss, & Rosenberg,
2008). For instance, Anderson and Hoffmeister (2007) found that the development of a PD
course that addressed mathematics content for middle school mathematics teachers helped
to increase their content knowledge and conceptual understanding of mathematical
concepts based on pre- and post-tests. Also, Feuerborn, Chinn, and Morlan (2009) reported
that teachers’ CK significantly improved after participating in a week-long professional
development course emphasizing algebra concepts, probability, and geometry concepts.
Although PD is a general term for a variety of activities and interactions designed to
improve teacher practices, there are several core features of effective PD programs
(Desimone, 2011; Garet, Porter, Desimone, Birman, & Yoon, 2001; Leko & Brownell, 2009).
Specifically, effective PD should be: (a) content focused and address knowledge and
pedagogy specific to the targeted subject matter (Garet et al., 2001; Yoon, Duncan, Lee,
Scarloss, & Shapley, 2007); (b) coherent and aligned to teacher needs, school-wide goals,
and curriculum standards (Leko & Brownell, 2009); (c) collaborative, allowing teachers to
discuss their practices and work collectively to address student achievement (Garet et al.,
2001); and (d) active and engaging, affording teachers the opportunity to actively
participate in the PD session and practice what they learned (Desimone, Porter, Garet.
Yoon, & Birman, 2002). Therefore, when designing effective PD opportunities, it is
16
important to incorporate these features by focusing on specific content, addressing
identified teacher goals and needs, and promoting collaboration and active teacher
engagement. Given the increase of including students with disabilities in general education
classrooms, it is imperative that secondary general and special education teachers receive
professional development for improving their CK and PCK in mathematics as well as their
ability to meet the needs of a diverse classroom.
When designing PD opportunities, an important feature to consider is the format of
the PD experience. Emerging technologies, such as video and online instruction, make it
possible to enhance PD content, providing an advantage over more traditional instructor
directed PD formats (Frey, 2009). One particularly promising example is using computer
animated representations which involve computer generated representations of actual
videotaped classroom interactions which can provide an advantage over more traditional
PD formats (Moreno & Ortegano-Layne, 2008). Specifically, the researchers found that use
of video animations and narrative exemplars (written accounts of teacher-student
interactions during instruction) are a “promising tool for instruction and research” (p.
463), and reported that animations offer greater flexibility for aligning instructional
objectives and provide an efficient means for capturing a variety of learning situations. As a
joint program between the University of Maryland College Park and the University of
Michigan under a National Science Foundation (NSF) funded research and development
project, the Thought Experiments in Mathematics Teaching (ThEMaT) project created
animated representations of mathematics teaching based on models of content specific
classroom interactions (Center for Math Education, n.d.). These animations are also being
considered for use in teacher education courses as a tool for increasing teacher CK as well
17
as for analyzing student learning. In a study of 21 inservice and preservice teachers, Moore-
Russo and Viglietti (2010) analyzed teachers’ interpretations of and responses to a set of
animated representations of geometry content created under the ThEMaT project. Results
from this study indicated that teachers found the animations useful for considering the
effects of their pedagogical decisions on student learning and engagement and helped to
improve their content knowledge. However, there is currently no research on the use of
animations as a vehicle for helping teachers adapt their instruction to meet the needs of
students with disabilities or those experiencing mathematics difficulties, particularly in
geometry. Thus, future studies are needed to examine the effects of animated exemplars as
a means for increasing the content and pedagogical content knowledge of general and
special education as well as the impact on the teachers’ ability to adjust their instruction for
teaching in an inclusive setting.
Statement of Purpose
Improving the performance of secondary students in geometry is critical in order to
prepare students for success in post-secondary education, daily living skills, and career
attainment. However, many students, particularly students with LD struggle with learning
geometry content. Ensuring that all students receive quality instruction and have access to
the general education curriculum is critical for addressing the challenges and difficulties
that hinder the learning process for many students with disabilities. Given that a majority
of students with special needs are educated in the general education classroom, general
and special educators are being encouraged to collaborate, often teaching alongside one
another or being responsible for instruction to a diverse group of students. The self-efficacy
beliefs of teachers play a significant role in teacher’s ability to adapt their instruction to
18
meet the needs of a diverse student population. Research shows that general educators
generally feel underprepared to provide appropriate instruction for students with special
needs. Likewise, special education teachers report that they often lack the necessary
content knowledge and understanding needed to help students meet rigorous academic
standards. In order to ensure that all students improve their conceptual knowledge of
geometry content, teachers must have a deep understanding of geometry content
knowledge as well as strong pedagogical content knowledge.
Teacher inservice PD is an effective way to increase teacher CK and PCK. Although
using emerging technologies, such as computer animated representations of teaching, can
be an effective tool for facilitating teacher PD, the research base for using such animations
as a means for increasing teacher CK and the use of effective instructional practices is
limited. Further, there is no research on the use of emerging technologies in PD for general
and special education teachers of students with disabilities. Thus, the purpose of this study
is to extend a growing body of PD research involving use of animated representations for
increasing teacher’s self-efficacy, content knowledge, and pedagogical content knowledge
of geometry topics involving congruence and similarity for inservice special education and
general education mathematics teachers within an inclusive setting.
Research Questions and Hypotheses
The current study is guided by the following research questions:
1. What is the effect of a teacher professional development course involving use of
animated representations addressing congruence and similarity on special
education and mathematics education teacher self-efficacy, content knowledge, and
pedagogical content knowledge in an inclusion setting?
19
2. How do general education and special education teachers compare in their
understanding of congruence and similarity as well as their self-efficacy for teaching
geometry in an inclusive setting?
3. To what extent do general and special education teachers find a blended learning
professional development course as an effective means for increasing teacher
content and pedagogical content knowledge in geometry?
The hypotheses for this study are:
1. It is predicted that the PD course will be effective for increasing teacher content
knowledge, pedagogical content knowledge, and self-efficacy.
2. It is anticipated that general education teachers will demonstrate a greater self-
efficacy for teaching geometry and special education teachers will have a greater
self-efficacy for teaching in an inclusive setting.
3. It is predicted that general education teachers will report a greater increase in self-
efficacy for teaching in an inclusive setting compared to special education teachers;
4. It is predicted that special education teachers will report a greater increase in
content and pedagogical content knowledge in geometry compared to general
education teachers.
20
Definition of Key Terms
Animated Representations – manufactured episodes of classroom interactions whereby the
classroom instruction is depicted in detail with vague details of the setting and
individuals (Herbst & Miyakawa, 2008).
Congruence – the concept in geometry that two shapes are equivalent by a combination of
translations, rotations, and/or reflections (Seago, Jacobs, & Driscoll, 2010)
Content Knowledge – the understanding of mathematical facts and concepts used to carry out
mathematics instruction (Krauss et al., 2008).
Inclusive Setting – the learning environment where students with disabilities meaningfully
participate in the general education classroom with their non-disabled peers (Bateman
& Bateman, 2002).
Learning Disability – a disorder in one or more of the basic psychological processes involved in
understanding or in using language, which may manifest itself in the imperfect ability to
listen, think, speak, read, write, spell, or do mathematical calculations (IDEA, 2004).
Pedagogical Content Knowledge – teacher’s ability to present mathematical knowledge in an
effective way for students to learn (Shulman, 1986).
Professional Development – a range of formal and informal learning activities or experiences
intended to increase teachers’ knowledge and skills, improve their practice, and
contribute to their personal, social, and emotional growth (Cohen, McLaughlin, and
Talbert 1993).
Self-efficacy – an individual’s perception of their ability to organize and execute actions that
will bring about desired results (Bandura, 1977).
21
Similarity – the concept in geometry that states that two figures are similar if one is congruent
to a dilation of the other (Seago, Jacobs, & Driscoll, 2010).
Universal Design for Learning – an educational framework used in helping to improve
educational outcomes through targeting support to individuals and by providing
multiple methods for instruction, assessment and engagement (CAST, 2006).
22
Chapter 2: Review of the Literature
Achieving mathematical literacy is significant for students to realize success
academically as well as increase their opportunities to obtain meaningful employment.
Students who participate in higher level mathematics courses in high school are more
likely to have a higher income, use new technology, vote, and engage in civic leadership
(National Mathematics Advisory Panel [NMAP], 2008). Geometry, in particular, is a critical
strand of the school mathematics curriculum (Cawley, Foley, & Hayes, 2009). Although
there is less emphasis within the secondary mathematics curriculum, as compared to
algebra and number sense concepts, learning geometry concepts offers students numerous
opportunities to improve their cognitive ability, as well as reason and communicate about
mathematics in multiple ways (Seago, Driscoll, & Jacobs, 2010). Understanding congruence
and similarity in particular connects a variety of important mathematical topics including
linear functions, transformations, proportional reasoning, and modeling (Seago, Jacobs, &
Driscoll, 2010).
Due to recent legislation, such as No Child Left Behind (NCLB) and the Individuals
with Disabilities Education Improvement Act (IDEIA), as well as the increase in
inclusionary practices, a majority of students with learning disabilities (LD) take their
geometry course in the general education classroom (Steele, 2010). A major premise of the
1997 and 2004 Reauthorizations of IDEIA is to ensure that students with disabilities have
access to grade appropriate curriculum and requires schools to educate this student
population with their non-disabled peers to the greatest extent possible (IDEIA, 2004). In
compliance with NCLB, all students are assessed in mathematics annually in grades 3-8,
and at least once more between grades 10 – 12. These assessments must align with the
23
state’s academic content and student academic achievement standards and must assess
higher-order thinking skills and understanding. Results from assessments are reported
with disaggregation of scores for students with disabilities, therefore, students with
disabilities are being held accountable for the general education secondary mathematics
content. As such, 62% of secondary students with LD participate in mathematics courses in
the general education classroom (Newman, 2006); however, only 13.6% of students with
LD perform on grade level during secondary school (Wagner et al., 2003).
In this chapter, I will first summarize the achievement results of U.S. students in
mathematics, particularly geometry. I then discuss the characteristics of students with
learning disabilities and the role of education reform, including policies for including
students with disabilities in the general education classroom. This is followed by a review
of the literature, that supports the design of this study. Specifically, the literature review
begins with an overview of the research on teacher self-efficacy, content knowledge and
pedagogical content knowledge in mathematics, followed by a review of the current
literature of professional development as it relates to increasing instruction and learning in
mathematics. I also examine the application and use of animated representations of
teaching in mathematics professional development. The chapter concludes with a summary
of findings from the literature and implications for the current study.
Proficiency of U.S. Students in Mathematics
In general, students in the U.S. have not performed well on assessments of their
understanding of geometry concepts. Data from the Trends in International Mathematics
and Science Study (TIMSS) and the Program for International Student Assessment (PISA)
indicate that students in the U.S. are performing below the level of many other
24
industrialized countries in the skills that are critical for understanding geometry concepts
such as their ability to use visual and spatial skills to solve problems, make generalizations,
and organize information to complete sequential tasks (Mullis et al, 2012). U.S. students
have also not made the achievement gains expected on national assessments. On the
geometry portion of the National Assessment of Educational Progress (NAEP), the average
scale score was 17 percentage points below the proficient level for eighth grade students
and 21 percentage points below the proficient level for twelfth grade students (National
Center for Education Statistics, 2013). The average scale score for eighth and twelfth grade
students with disabilities on the geometry portion of the assessment was 31 and 38 points
lower, respectively, than their general education peers (National Center for Education
Statistics, 2013).
Characteristics of Students with Learning Disabilities
Certain characteristics of students with LD may impede their geometry
performance, including difficulties with calculation fluency and procedural knowledge
(Flores, Houchins, & Shippen, 2006; Garnett, 1998; Geary, 2004). In addition, students with
disabilities in mathematics tend to lack an understanding of mathematical concepts, which
hinders their problem solving and reasoning abilities (Miller & Mercer, 1997; Parmar,
Cawley, & Frazita, 1996). Poor strategy knowledge and strategy use also contribute to
difficulties across mathematics achievement (Flores, Houchins, & Shippen, 2006; Montague
& van Garderen, 2003). Further, they may experience poor self-esteem, decreased
motivation, and/or passivity in the classroom (Gagnon & Maccini, 2001).
Mathematics Reform and the Role of Inclusion
25
In response to the lagging student achievement in mathematics, the National Council
of Teachers of Mathematics (NCTM, 2000) addressed the need for more rigorous standards
by publishing the Principles and Standards for School Mathematics. The NCTM Standards
focus on conceptual understanding and real-world problem solving and reflect a belief in
the importance of mathematics for all students, including students with disabilities. The
Content Standards describe the content that all students should learn from pre-
kindergarten through twelfth grade (i.e., number and operations, algebra, geometry,
measurement, data analysis, and probability). The Process Standards (i.e. problem solving,
reasoning and proof, communication, connections, and representations) describe ways of
learning the content knowledge. Additionally, the high school mathematics curriculum
should emphasize mathematical reasoning (i.e. drawing conclusions based on evidence)
and sense making (i.e. developing an understanding of a concept by connecting it to
existing knowledge) in all courses, for students of varying abilities (NCTM, 2009).
Findings from research studies of mathematics education programs in high-
performing countries and suggestions from organizations such as NCTM, NCR, and ADP,
pointed to the need for more focused and coherent mathematics standards in the U.S.
Together, the National Governors Association (NGA) and the Council of Chief State School
Officers (CCSSO) collaborated to create a national set of standards in both language arts
and mathematics called the Common Core States Standards (CCSS, 2010). The CCSS in
mathematics focus on understanding key concepts by reviewing and building on the
organizing principles of mathematics and how the properties of operations lead up to more
advanced concepts (CCSS, 2010). The CCSS for mathematics also provide a set of practices
that students should demonstrate to show depth and breadth of mathematical knowledge.
26
The sections below describe the eight Common Core Standards for Mathematical Practice
and how each relates to geometry instruction.
Make sense of problems and persevere in solving them. This practice standard
involves students performing actions such as explaining the meaning of a problem to
themselves, looking for entry points, and making conjectures and a plan to solve a problem.
This standard also includes considering analogous problems, monitoring and evaluating
progress, checking answers with alternative methods and generally insuring if steps taken
or derived answers make sense in the context of the problem (CCSS, 2010). For example,
students may be asked to develop a congruence statement explaining how two shapes or
objects are related to each other. Students will identify the corresponding parts that match
each other as well as map certain transformations to prove their statement.
Reason abstractly and quantitatively. This practice standard involves students
demonstrating the ability to create coherent representations of problems, attend to the
meaning of quantities, use properties of operations and objects, decontextualize situations
by manipulating mathematical symbols abstractly without necessarily attending to the
original referents, and to pause and contextualize the symbols during the manipulation to
make sense of the symbols concretely in terms of the original context (CCSS, 2010). Babai
et al (2009) investigated an intervention for helping students solve geometric tasks by
raising student awareness to relevant information. The results of the study demonstrated
that students improved their fluency and accuracy for solving geometry problems by
attending to the relevant variables and overcoming irrelevant information presented in the
tasks. This is an example of students’ ability to reason abstractly and quantitatively.
27
Construct viable arguments and critique the reasoning of others. Students who
demonstrate this standard are able to understand stated assumptions/definitions, make
conjectures, build progressions of statements to explore the truth of conjectures, justify
conclusions, compare the effectiveness of two plausible arguments, and distinguish and use
correct logic and reasoning. They are also able to communicate findings, respond to the
arguments of others, decide if an argument makes sense, and ask clarifying questions
(CCSS, 2010). Problem posing activities is a way for students to construct reasonable
arguments and make conjectures about geometry concepts. Problem posing activities
involve the generation of new problems and questions aimed at exploring a given situation,
as well as the reformulation of given problems (Silver, 1994). For example, students may
generate new problems when formulating a geometric proof. Providing students with
opportunities to pose their own problems can foster more diverse and flexible thinking,
enhance students’ problem solving skills, broaden their perception of mathematics, and
enrich and consolidate their knowledge of basic concepts (Brown & Walter, 1993; English,
1996).
Model with mathematics. This practice standard involves students applying
mathematics to solve everyday problems, using multiple representations such as graphs,
tables and equations and interpreting/reflecting on the mathematical results in the context
of the situation (CCSS, 2010). Cihak and Bowlin (2009) examined the effect of students
using video modeling to solve geometry problems. Teachers created video clips of step-by-
step processes for solving problems, which were linked to handheld devices used by three
students diagnosed with LD. The researchers reported that each participant acquired and
maintained new geometry skills.
28
Use appropriate tools strategically. This practice standard involves students
considering and using available tools (e.g., concrete model, ruler, compass, calculator,
spreadsheets, dynamic geometry software) and evaluating their effectiveness for a given
situation to help students explore and deepen their understanding of mathematical
concepts (CCSS, 2010). A recent study that was conducted on this practice was by Guven
(2012), who examined the effect of Dynamic Geometry Software (DGS) on students’
learning of transformation geometry. Study participants were 68 eighth grade students
(36 in the experimental group and 32 in the control group). Students were taught how to
use the Cabri software and used the DGS to explore and test the characteristics of
transformations. Students were also able to check their understanding by completing
practice problems. Pre- and post-test results demonstrated that the experimental group
outperformed the control group in terms of academic achievement and the level of
understanding of transformation geometry.
Attend to precision. This practice standard involves students using clear and
precise definitions in discussions and explanations of their own work, stating the meaning
of and/or correctly using symbols specifying units of measure, calculating with precision
that is appropriate for a problem context, and giving carefully formulated explanations to
each other (CCSS, 2010). Poon (2011) report the use of a study of students using a simple
trigonometric problem that produces several answers depending on the method used to
solve the problem. Through this activity, students learned the importance of having
precise diagrams when solving geometric problems.
Look for and make use of structure. This practice standard involves students
analyzing objects and examples to discern patterns and structures, extending lines in
29
geometric figures to assist in problem solving, and “seeing” complicated things such as an
algebraic expression as either a single whole or composed of several objects (CCSS, 2010).
For example, students may be required to analyze and construct the relationship between
three-dimensional geometric figures and the related two-dimensional representations to
solve problems.
Look for and express regularity in repeated reasoning. This practice standard
involves students noticing repeated calculations and looking for general methods or
shortcuts, maintaining oversight of the problem solving process, and continually evaluating
reasonableness of approach and results (CCSS, 2010). For example, students may use
ratios to solve problems involving similar figures. Another example is for students to
describe the effect on area, perimeter and volume when or more of the dimensions are
changed.
The standards and practices inform what and how teachers should instruct students
for learning mathematical skills and concepts. While the Geometry standards are outlined
through all grade levels beginning with kindergarten, students at the secondary level begin
to formalize their geometry experiences from the earlier grades, using more precise
definitions and developing careful proofs. The concepts of congruence and similarity, in
particular, are understood through geometric transformations, which is consistent with the
NCTM Principles and Standards for Mathematics (CCSSI, 2010; NCTM 2000). Thus the
CCSS are important for considering helping teachers meet the needs of all students.
In addition to the NCTM and Common Core standards, reports from the National
Mathematics Advisory Panel (NMAP; 2008) and the American Diploma Project (ADP, 2004)
emphasize a rigorous curriculum for all learners. Authors of both reports suggest
30
mathematics benchmarks, which include foundational skills to prepare elementary age
learners for algebra (i.e., fluency and conceptual understanding of whole numbers,
fractions, and certain aspects of geometry; NMAP, 2008) and algebra skills necessary for
completion of Algebra during secondary education (i.e., linear equations, quadratic
equations, functions, and polynomials; ADP, 2004; NMAP, 2008). Considering that the No
Child Left Behind (NCLB) Act of 2001 mandates that all students, with few exceptions,
master the general education curriculum, participate in standardized assessments, and
achieve passing levels of performance, it becomes even more imperative to study the
effectiveness of inclusion programs from a variety of perspectives. Furthermore,
proportionately, students with LD are the largest special education group to be included in
general education classes. Forty-nine percent of students classified with specific LD spent
80% or more of each school day in a general education classroom. These students are not
among the groups exempt from state and national standardized tests (U.S. Department of
Education, 2003).
Co-teaching
Shifts in policy, which place an emphasis on educating students with disabilities in
the general education classroom, have increased the need for modified forms of instruction,
such as co-teaching (Palmer, 2005). Co-teaching is a service delivery model in which two or
more professionals share responsibility for a group students in the same workspace
(Murawski, 2005). The history of co-teaching can be traced back to the 1960’s, when the
concept was popularized as an example of progressive education (Villa, Thousand & Nevin,
2008). Co-teaching was further advanced in the 1970’s by legislated school reforms and
teachers’ increasing need to modify instruction for a more diverse student population.
31
Today, co-teaching is highly regarded as a primary answer to federal legislative changes to
NCLB and IDEA, as well as questions arising from the challenges dealing with the increase
in students requiring special education services (Villa, Thousand, & Nevin, 2008).
Research into the effectiveness of co-teaching has been limited and demonstrated
mixed results. Researchers Wendy Murawski and H. Lee Swanson (2001) examined data-
based studies on the effectiveness of co-teaching. The report found that only six of 89
reviewed articles provided sufficient quantitative information for an effect size to be
calculated. Special education students were primarily students with learning disabilities
and/or low achievement and the studies encompassed all grade levels. Walther-Thomas
(1997) did a study on co-teaching models in 23 schools across eight school districts.
Positive outcomes were documented including improved academic and social skills of low-
achieving students, improved attitudes and self-concepts reported by students with
disabilities, and more positive peer relationships. Students perceived that these
improvements were the result of more teacher time and attention (Walther-Thomas,1997).
The co-teachers also reported gains in professional growth, personal support, and
enhanced sense of community within the general education classrooms. However, the most
frequently mentioned drawback was the lack of staff development to learn how to be more
effective teachers.
Research suggests that one of the greatest issues faced by co-teachers has to do with
content knowledge and expertise (Mastropieri, Scruggs, & Graetz, 2005). Special education
teachers at the secondary level tend to be trained in learning differences and
accommodations but not content mastery, while their general education counterparts are
typically trained in content mastery at a high level. This study will address this particular
32
concern by providing professional development to special education and general education
co-teachers in geometry to increase content knowledge as well their efficacy for teaching in
an inclusion setting.
Universal Design for Learning
Universal design for learning is an educational framework used in helping to
improve educational outcomes through targeting support to individuals by reducing
barriers to learning (Rose & Meyer, 2002). UDL expresses the idea that standard
educational goals, materials, methods, and assessments may cause barriers to learning or
hinder students from accessing the curriculum (CAST, 2009; Edyburn, 2010). For example,
print resources (such as mathematics textbooks) contain mathematical terminology that
may be challenging for students with reading difficulties. Providing alternative ways to
present information (i.e. virtual manipulatives) may help students overcome their reading
challenges, while allowing them to access the curriculum. The organization of the UDL
framework focuses on three cognitive networks: (a) recognition network, (b) strategic
network, and (c) affective network (CAST, 2009). The recognition network is the what of
learning, which occurs in the occipital lobe or back part of the brain (CAST, 2009; Jensen,
2000; Restak, 2004). The type of knowledge dealing with applications is declarative
knowledge, in which learners identify and interpret patterns using their five senses. The
strategic network is the how of learning (CAST, 2009). The type of knowledge applied is
procedural knowledge in which learners plan, execute, and monitor their actions and skills.
The affective network is the why of learning and involves the application of affective
knowledge in which learners evaluate and set priorities using their emotions and their
interest for learning (CAST, 2009; Gordon, n.d.). Through strategic planning and training
33
using the UDL teaching principles, teachers become engaged in how to include the learning
modalities of their students, regardless of the student’s cognitive development and
personal interests (CAST, 2009; Furner, Yahya, & Duffy, 2005; Rose & Meyer, 2002).
Learning modalities are the ways in which learners process information into
memory from learning by seeing, learning by hearing, and learning by doing (CAST, 2009).
The recognition principle applies to presenting information in multiple formats (CAST,
2009). The strategic principle develops the idea that multiple pathways are available for
learners to conceptualize and demonstrate what they needed to learn (CAST, 2009). The
affective principle provides multiple ways for learners to engage in real life learning
experiences using emotions (CAST, 2009). By applying these principles, teachers can
effectively plan and implement instruction for all learners.
Current trends and policies support the adoption of inclusion or the practice of
educating students with disabilities in the general education setting with their non-
disabled peers (Idol, 2006). Inclusive practices allow students with special learning and/or
behavioral needs to have access to the general education curriculum, which supports IDEA
mandates for educating students with disabilities in the least restrictive environment
(IDEA, 2004). However, inclusion often poses a challenge for general education teachers
who often feel ill-prepared to teach students with LD (Borasi, Fonzi, Smith, & Rose, 1999;
Jenkins & Ornelles, 2009; Maccini & Gagnon, 2007). In a recent study of seven in-service
middle school mathematics teachers (Desimone & Parmar, 2006), each teacher reported
that they felt that their undergraduate and graduate teaching programs did not adequately
prepare them to teach mathematics in an inclusive classroom. These results are consistent
with a study conducted by Rao and Lim (1999) in which general education teachers
34
reported that they needed, among other supports, training on curriculum adaptations and
instructional strategies to be successful in an inclusion classroom. Due to inadequate
teacher training and relevant PD for teaching students with LD in mathematics, general
education teachers may lack confidence in their ability to provide accommodations for
those students (Jordan, Stanovich, & Roach, 1997). Further, special education teachers, who
receive training for meeting the needs of students with LD, may feel inadequate with
regard to their knowledge of the subject area content (Leko & Brownell, 2009). For
instance, in a study of 34 beginning special education teachers (less than 3 years teaching
experience), Brownell et al (2009) found that special education teachers struggled with
domain specific content and pedagogical knowledge and, instead, relied more on general
classroom management practices when providing instruction. The implications of these
results suggest that special education teachers may rely on general instruction practices,
which may limit their ability to provide quality instruction to their students.
Frey (2009) reported that the shortage of special education teachers also
contributed to the lack of knowledge in the profession. The increased demand for qualified
special education teachers often requires schools to hire teachers who are less trained and
experienced to work with students with disabilities. Special education teachers completing
short, intensive alternative training programs for certification may find it difficult to help
general education teachers meet the needs of students with diverse learning needs.
General and special education teachers need quality preparation and ongoing training to
increase their ability to provide instruction to all students.
Teacher Self-Efficacy
In order to meet the diverse learning needs of students, teachers must believe in
35
their own ability to do so. Self-efficacy, in regards to education, plays a critical role in the
instructional decisions teachers make on a consistent basis (Kosko & Wilkins, 2009).
Bandura (1986) defined self-efficacy as, “beliefs in one’s capabilities to organize and
execute the courses of action required to produce given attainments” (p. 2). Bandura
postulated that the theory of self-efficacy contains two expectancies, self-efficacy and
outcome efficacy. Self-efficacy expectation provides individuals a way to decide whether
they have the ability to perform the required task at the desired level of competency,
whereas outcome expectancy provides individuals a way to decide if they have
accomplished a task at a desired level (Hoy & Woolfolk, 1993).
With regard to education research, Ashton and Webb (1986) focused the self-
efficacy construct on teachers by defining teachers’ sense of self-efficacy as a “situation-
specific expectancy” that they can affect or influence student learning. Similar to Bandura
(1986), Ashton and Webb recognized that these beliefs affect a teacher’s choice of
classroom activities, the amount of effort the teacher is willing to expend, and his or her
persistence when experiencing difficulties. When teachers believe in their ability to meet
the learning needs of their students, they design and deliver instruction which provides
students access to the skills and concepts they are learning while enabling them to
construct new knowledge and understanding (Ashton & Webb, 1986).
Teacher efficacy is a self-perception and not necessarily an outcome of teaching
effectiveness (Ross & Bruce, 2007). However, researchers have linked teachers’ self-
efficacy to a variety of positive outcomes. For example, high school teacher self-efficacy has
been positively correlated with improved student achievement (Ashton & Webb, 1986;
Dembo & Gibson, 1985; Tschannen-Moran, Hoy, & Hoy, 1998). Researchers claim that
36
teacher efficacy relates to student achievement as it results in teachers’ efforts to adapt
instructional practices that support student learning (Allinder, 1995; Ashton & Webb,
1986; Caprara, Barbaranelli, Steca, & Malone, 2006; Dembo & Gibson, 1985). In addition,
teachers with higher efficacy levels are more apt to plan engaging lessons and interact with
students to encourage their participation in the lesson (Schunk, 2008). In regards to
teachers working in inclusive settings, teachers with high levels of efficacy are willing to
involve students with disabilities in class discussions and persist in educating them
(Brownell & Pajares, 1996; Nunn, Jantz, & Butikofer, 2009), while maintaining best
practices for classroom management (Woolfson & Brady, 2009). Teachers with higher
efficacy also tend to provide increased support to struggling students in their classes such
as one-one guidance and additional opportunities for practice (Dembo & Gibson, 1985).
Recent research has further defined Bandura’s efficacy theory in the context of
education. Individual efficacy and collective efficacy are two independent measures often
used in research regarding teacher efficacy (Bandura, 1998; Tschannen-Moran, Hoy, & Hoy,
1998). Although individual efficacy is generally defined as a teacher’s belief in his or her
skills and abilities to positively impact student achievement, collective efficacy describes
the teacher’s belief about his or her colleagues’ effectiveness to positively impact student
achievement (Zambo & Zambo, 2008). Given that personal teaching efficacy is influenced
by the additional education teachers receive (Hoy & Woolfolk, 1993), general educators
who are faced with the challenges of having students with varying levels of ability and
behavioral responses in their classrooms should be provided the knowledge to understand
the needs of such students and the skills to teach these students in their classrooms.
Therefore, future research should focus on the needs of general and special education
37
teachers who teach in an inclusive setting.
Teacher PD can influence self-efficacy when the knowledge and tasks that are
covered are relevant and connected to the teacher’s classroom situation (McLaughlin &
Berman, 1977; Scribner, 1998). In a study of 12 third to sixth grade teachers who
participated in an intensive PD program, researchers Bruce and Ross (2008) found that the
participants shifted to more standards based instruction, resulting in higher teacher self-
efficacy for facilitating student discussion and improving the quality of the tasks assigned.
The PD program spanned six months and consisted of four sessions that involved
collaborative lesson planning, examining student work samples, and modeling effective
teacher practices. In a more extensive study (Powell-Moman & Brown-Schild, 2011),
researchers investigated the impact of a two-year professional development program on
teachers’ self-efficacy for adopting inquiry-based instruction (e.g., cognitive processes of
scientists). Participants were 23 practicing science and mathematics teachers at the
elementary and secondary levels who participated in a fellowship program that used a
scientist-teacher partnership to help teachers use inquiry-based instruction as well as
develop their teacher leadership skills. Results from this study support prior research for
using PD to increase teacher knowledge and self-efficacy.
Content and Pedagogical Content Knowledge
The literature on the content and pedagogical content that mathematics teachers
need to know is substantial. Over the past two decades, policymakers and national
organizations, such as the National Council of Teachers of Mathematics (NCTM), have
called for improvements to the teaching and learning of mathematics. They argue that
traditional, procedural-based approaches to mathematics teaching do not prepare enough
38
students for higher-level mathematics. Success in higher-level mathematics is dependent
on students’ understanding of concepts as well as procedures. In order for students to
receive mathematics instruction that attends to concepts as well as procedures, teachers
must receive additional training and support to deliver instruction in a more conceptually
based way (National Council of Teachers of Mathematics [NCTM], 1989; 1991; 2000;
Kilpatrick et al., 2001).
Though mathematicians and mathematics education experts may differ over the
relative emphasis of particular mathematics concepts, they agree that conceptual
understanding is a necessary condition for effective teaching. Teachers need to know why,
not just how, so that they can deliver meaningful instruction to students. Teachers need to
be able to make connections among mathematical concepts so that students obtain a
coherent, rather than a fragmented, view of mathematics (National Math Panel Advisory
Panel [NMAP], 2008). Further, in order for teachers to help students demonstrate the CCSS
Standards for Mathematical Practice, teachers must have a deep understanding of the
mathematics concepts.
Related to strong conceptual understanding of the topics they teach, experts also
agree that teachers need pedagogical content knowledge, which is knowledge about how to
teach the concepts in a way that is effective for students to understand (Shulman, 1986).
For example, a teacher with conceptual understanding about the meaning of fractions
might use that knowledge to improve the precision or coherence of an explanation about
fractions.
Pedagogical content knowledge is viewed as a set of attributes that facilitate the
transfer of the teacher’s knowledge of the content to their students (Geddis, 1993).
39
Shulman (1987) further posited that these attributes that the teacher possesses allows
presentation of content in a way that is personally meaningful for students. This includes
what makes learning geometry concepts, for example, particularly more accessible or
challenging for students with diverse learning needs.
Organization of the Review of the Literature on Professional Development
In this section, I present a comprehensive review of current research involving
professional development studies on teachers’ self-efficacy, content, and pedagogical
content knowledge in geometry. Specifically, I examine the salient features of professional
development models that are effective for increasing teachers’ self-efficacy as well as their
content and pedagogical content knowledge in geometry.
Search Procedures
For this literature review, I conducted an electronic search using the following
databases: Education Research Premier (EBSCO); ERIC; PsychINFO; and Professional
Development Collection. Potentially relevant studies published between 1989 and 2012
were identified using these descriptors in varying combinations: teacher self-efficacy;
inclusion; math*; geometry; professional development; content knowledge; and pedagogical
content knowledge. This initial search yielded a total of 116 studies. Selected studies were
narrowed for this review using the following criteria: (a) published in a peer-reviewed
journal; (b) included general and/or special education teachers, and (c) either examined
the effects of a PD program on teacher self-efficacy in mathematics or studied the effects of
a PD program on teachers’ content and pedagogical content knowledge in geometry.
Secondary was defined as students in grades 6-12 or approximately ages 11-18. Applying
these conditions, 10 studies met the criteria for inclusion (Arbaugh, 2003; Borasi, Fonzi,
40
Smith, & Rose, 1999; Brown & Benken, 2009; Feuerborn, Chinn, & Morlan, 2009; Kimmel &
Deek, 1999; Merrill, Devine, & Brown, 2010; Powell-Moman & Brown-Schild, 2011; Ross &
Bruce, 2007; Swackhamer, Koellner, Basile, & Kimbrough, 2009; Zambo & Zambo, 2008).
Overview of the Studies
A summary of each of the studies is presented in Table 1. Overall there were 476
teacher participants, with 469 general education math teachers and 7 special education
teachers. Two of the ten studies reported the gender of the participants, which totaled 6
female and 4 male teachers (Arbaugh, 2003; Brown & Benken, 2009). A third study
reported that teacher participants were mostly female (Feuerborn et al, 2009). Six of the
ten studies feature PD programs aimed at increasing teacher content and pedagogical
content knowledge in geometry (Arbaugh, 2003; Borasi et al., 1999; Brown & Benken,
2009; Kimmel & Deek, 1999; Merrill et al., 2010; Powell-Moman et al., 2011), two studies
focus on increasing teacher knowledge and self-efficacy (Feuerborn et al., 2009;
Swackhamer et al., 2009), and two studies address the effects of a PD program on teacher
efficacy alone (Ross & Bruce, 2007; Zambo & Zambo, 2008). Five of the ten studies
(Arbaugh, 2003; Borasi et al., 1999; Brown & Benken, 2009; Kimmel & Deek, 1999; Merrill
et al., 2010), are qualitative in nature, (Arbaugh, 2003; Borasi et al., 1999; Brown &
Benken, 2009; Kimmel & Deek, 1999; Merrill et al., 2010), three studies (Powell-Moman et
al., 2011; Swackhamer et al., 2009; Zambo & Zambo, 2008) used correlational analysis
(Powell-Moman et al., 2011; Swackhamer et al., 2009; Zambo & Zambo, 2008), one study
(Ross & Bruce, 2007) used a randomized field trial, and the remaining while another study
(Feuerborn et al., 2009) used a mix of qualitative and quantitative methods. (Feuerborn et
al., 2009). Eight of the ten studies included only general education mathematics teachers
41
(Arbaugh, 2003; Brown & Benken, 2009; Feuerborn et al., 2009; Merrill et al., 2010; Powell-
Moman & Brown-Schild, 2011; Ross & Bruce, 2007; Swackhamer et al., 2009; Zambo &
Zambo, 2008). The remaining two studies included both general and special education
teachers (Borasi et al., 1999; Kimmel & Deek, 1999).
Content Analysis
Since professional development is a broad term that encompasses a variety of
structures and delivery types, it is useful to frame a discussion about the more effective
types of professional development around a few, core structural features that apply to
most, if not all, professional development models. The professional development literature
identifies specific features of professional development models that are deemed to be more
promising or effective for improving teacher learning (Garet et al., 2001; Hawley & Valli,
1999). As part of a nationally representative sample of more than 1,000 teachers who
participated in professional development (mostly mathematics and science), researchers
identified several critical features that contextualize professional development (Birman et
al., 2000). These features included the form or type of PD activity, the duration of the PD
program, and how the participants were selected and arranged. In this section, I discuss the
selected studies in regards to four features that have been deemed important for evaluating
PD programs: 1) source; 2) purpose; 3) format; and 4) duration (Birman et al., 2000; Garet
et al., 2001). Each feature is defined, including best practices according to the literature,
followed by an analysis of the selected studies. The section is concluded with a summary of
the implications for the current study.
Source. The source, or origin of the PD program represents why the participants
engaged in the PD activity. For example, did the teachers voluntarily participate in the PD
42
or was it required by school administration or district. In general, professional
development activities can be described as more or less teacher-initiated, which is an
important distinction to make when categorizing the level of teacher ownership of learning
opportunities. In their review of the literature on effective types of professional
development, Hawley and Valli (1999) suggested that effective professional development
programs, to the greatest extent possible, involve teachers in the identification and
development of what they need to learn and how they can learn it. It is also important that
PD programs are connected to school and/or district-wide goals (Little, 1993).
With regard to the selected studies, seven of the ten reported that the participants
engaged in the PD program on a voluntary basis (Arbaugh, 2003; Brown & Benken, 2009;
Feuerborn et al., 2009; Merrill et al., 2010; Powell et al., 2011; Swackhamer et al., 2009; and
Zambo & Zambo, 2008). Generally, involving teachers in the process of selecting the PD
activity increases their sense of ownership and the amount of motivation and effort they
bring to learning situation (Hawley & Valley, 1999). Brown and Benken (2009) reported on
a PD program that offered two courses in content and pedagogy that were designed based
on the participant’s responses to surveys, current classes taught, and the curriculum
participants used. Although offering the PD on a voluntary basis increases teacher
participation, researchers also employ other methods. For example, Feuerborn et al. (2009)
reported that the participating teachers in the PD institute received incentives and
continuing education credits for completing the program. Research also supports the use of
school district and university partnerships as a means for increasing teacher knowledge in
a targeted subject area. Merrill et al. (2010) investigated the influence of a PD partnership
between the Peoria Public School District and Illinois State University. Eight mathematics
43
teachers participated in the partnership aimed at improving teachers’ trigonometric and
geometric knowledge. In another example, researchers Powell-Moman and Brown-Schild
(2011) reported on a study of the Kenan Fellows Program, an intensive PD program for
science, technology, engineering, and mathematics teachers. Participating teachers were
required to apply for acceptance into the program. Selected teachers were trained to
increase their responsibilities at their respective schools, deliver PD, and received six
graduate level credits.
Overall, this literature supports that effective professional development models
provide opportunities for teachers to be active participants in the selection of the types of
PD offered to allow for greater participation and learning engagement. Providing
incentives, such as course credits or teacher leadership opportunities are also effective
ways for increasing teacher buy-in and completion of PD programs that are involuntary.
While the proposed study will be voluntary, teacher participants will receive a modest
monetary incentive for their participation. In addition, the online activities will provide
teachers greater flexibility, which may enhance their participation by making them active
learners.
Purpose. The purpose of a professional development activity, which includes the
focus or emphasis of the PD, is another distinguishing characteristic of any professional
development activity or model. The purpose varies across activities and models. For
example, one might focus on deepening teachers’ understanding of the subject matter;
another might emphasize pedagogical skills; and another might focus on aligning teaching
practices with district standards and assessments.
Professional development activities that focus on deepening teachers’ content
44
and/or pedagogical content knowledge are thought to be more effective than other models.
For example, Cohen and Hill (1998) conducted a study of mathematics teachers in
California and found that student achievement was higher in schools where teachers had
participated in extensive, content-focused professional development. Garet et al. (2001),
utilizing a national probability sample of over 1,000 math and science teachers, came to a
similar conclusion. They found that content-focused professional development had a
significant positive direct effect on teacher self-reported knowledge and skills and a
significant positive indirect effect on changes in teacher practice. These findings are
consistent with Kennedy’s (1998) review of studies that linked various types of
professional development to student achievement and pedagogical content-focused
professional development had larger positive effects on student achievement – particularly
on students’ conceptual understanding – than more general types of professional
development. Publications by the National Council of Teachers of Mathematics (NCTM
1989; 1991; 2000) and the National Research Council (2001) also highlight the importance
of content and pedagogical content-focused professional development activities. Both
organizations recommend sustained, intensive professional learning opportunities for
teachers in these areas.
In a review of the selected studies, six of the ten featured PD programs that focused
on increasing teacher content and pedagogical content knowledge in geometry (Arbaugh,
2003; Borasi et al., 1999; Brown & Benken, 2009; Kimmel & Deek, 1999; Merrill et al.,
2010; Powell-Moman et al., 2011). For example, Merrill et al. (2010) developed a
partnership between Illinois State University and the Peoria Public School District to
provide a series of PD courses that aimed at improving teachers’ geometric and
45
trigonometric knowledge and their ability to teach mathematics using 3-D modeling
software. In another study, seven high school geometry teachers participated in PD study
groups that focused on developing a geometry curriculum that was more student-centered
and inquiry based (Arbaugh, 2003). The topics covered in the study groups included
triangle congruency, student reasoning, and engaging activities using The Geometer’s
Sketchpad.
This literature suggests that professional development models that focus on
building teachers’ content or pedagogical content knowledge are promising in terms of
promoting teacher learning that can improve the quality of instruction. The literature also
suggests that when the professional development is coherent with district standards and
assessments, it is likely to impact the quality of teaching and learning. Therefore, the
proposed study will focus on building and increasing teacher content and pedagogical
content knowledge in congruence and similarity concepts. These concepts are aligned with
the Common Core Standards and Practices for mathematics and teachers will be able to
apply their learning experience.
Format. The format, or how the PD is organized and delivered, can take many forms.
Garet et al. (2001) noted that the most widely used format, and the most evaluated in the
literature, is the traditional workshop, which typically occurs at structured times outside of
the classroom and is led by a content expert (Loucks-Horsley, Hewson, Love, & Stiles,
1998). Traditional workshop formats have generally been found to be ineffective for
increasing teacher knowledge and improving teacher practice (Garet et al., 2001; Loucks-
Horsley et al., 1998). Other formats include collaborative study groups, and individualized
coaching. For example, many teachers in Japan participate in lesson study groups that meet
46
regularly over the course of a school year to focus on improving a lesson or series of
interconnected lessons (Stigler & Hiebert, 1999). As a reference for this review, Garet et al.
(2001) coded PD formats into two broad categories: 1) traditional activities, which include
institutes, college courses, and out-of-district workshops or conferences, and 2) reform
activities (i.e., study groups, individualized coaching and mentoring, and school-university
partnerships)
In regards to studies selected for this review, six of the ten (Feuerborn et al., 2009;
Kimmel & Deek, 1999; Powell-Moman et al., 2011; Ross & Bruce, 2007; Swackhamer et al.,
2009; Zambo & Zambo, 2008) featured PD programs that fall under Garet et al.’s (2001)
characterization of traditional PD activities. For example, Zambo and Zambo (2008)
implemented a summer PD workshop led by an outside mathematics expert for 63 4th –
10th grade mathematics teachers. Similarly, Feuerborn et al. (2009) offered teachers two
five-day PD institutes that focused on mathematics content and pedagogy. The institutes
were led by an expert mathematics teacher employed by the participating school district.
Swackhamer et al. (2009) examined whether participation in a series of college level
courses increased the self-efficacy of secondary mathematics teachers. Kimmel & Deek
(1999) engaged general and special education teachers through inservice workshops and a
summer institute, which included a classroom practicum experience that included twelve
students with learning disabilities. The PD model included a focus on collaborative
practices, increasing teacher knowledge, and integrating teaching practices that address
the needs of special education students.
Four of the ten studies reviewed meet the Garet et al. (2001) criteria for reform PD
activities (Arbaugh, 2003; Borasi et al., 1999; Brown & Benken, 2009; Merrill et al., 2010).
47
Arbaugh (2003) examined the value of teacher participation in a school-based study group,
which was defined as a group of educators who meet on a regular basis to study varying
aspects of education. Specifically, seven geometry teachers met approximately once every
two weeks to discuss a range of topics related to geometry instruction and student
understanding. Analysis of teacher interviews and study group transcripts show that the
study groups had a positive impact on teacher efficacy as well as increased their content
knowledge. In another study, researchers developed a school-university partnership to
create a series of courses aimed at improving teacher content knowledge for implementing
three-dimensional modeling in the classroom. Results from formative and summative
evaluations conducted by the external evaluator indicated that teacher participants found
the program valuable for promoting innovative teaching strategies and for using practices
that lead to greater student understanding. Brown and Benken (2009) reported on a study
of a researcher and teacher collaboration as participants in two PD courses aimed at
increasing their content and pedagogical knowledge as well as building a community of
collaborative practice. Results from the study revealed that teacher educators must know
more about and be responsive to teacher needs and challenges before engaging them in PD.
Further, PD should provide a mechanism for participants to feel empowered to implement
change. In another study with middle school special education and general education
teaching teams, Borasi, Fonzi, Smith, and Rose (1999) found that a PD program that
involved engaging teachers in teaching and learning experiences for implementing an
inquiry approach to geometry instruction was effective for promoting reform based
instructional methods in mathematics as well as increasing teachers’ reflective practices. A
promising feature of this study is the benefit of collaboration among special and general
48
education teachers, which is significant for successful teaching in inclusive classrooms
(Buell, Hallam, & Gamel-McCormick, 1999).
The literature contains considerable support for professional development models
that promote collective and collegial participation among teachers. Garet et al. (2001)
found that collective participation had modest positive effects on the core features of
coherence and active learning. They defined collective participation as group participation
in professional development, such as participation by a department or grade-level group of
teachers. Talbert and McLaughlin (1994) found that teacher participation in active learning
communities enhanced professional knowledge and overall professionalism. Little (1993)
argued that teachers should have regular opportunities to engage intellectually with
colleagues both inside and outside of teaching. Thus, teacher learning communities should
include not only teachers within a department or within a school, but also content experts
and university researchers who are capable of infusing the learning communities with
relevant professional knowledge. The proposed study will allow general and special
education teachers to work together in face to face sessions as well as collaborate through
online discussions to promote collective participation and enhance the shared learning
experience.
Duration. Garet et al. (2001) defined duration as including both the total number of
teacher contact hours and the time span of the professional development. The researchers
found that both dimensions of duration had substantial positive direct effects on active
learning and coherence and modest positive effects on content focus. This finding suggests
that both “how much” and “how long” are important characteristics of professional
development. Yoon et al.’s (2007) meta-analysis of professional development and student
49
achievement found that the professional development models that showed gains in student
achievement averaged about 70 hours of professional development for one year.
For the studies selected for this review, only three of the ten reported PD activities
that incorporated over 70 hours of total duration (Kimmel & Deek, 1999; Merrill et al.,
2010; Powell-Moman & Brown-Schild, 2011). The general and special education teachers
who participated in the Kimmel and Deek (1999) study participated in a multiyear PD
program designed for teachers who work with student with disabilities. The program
consisted of a 3-week summer institute followed by 3-4 one-day follow up meetings during
the school year. The school-university partnership PD program used in the Merrill et al.
(2010) study featured over 80 hours of PD activities between the Summer 2009 semester
and the Spring 2010 semester. Participants in the Powell-Moman and Brown-Schild (2011)
study were part of a two-year PD institute made up of a four-week externship followed by a
two-week summer session during the first year, and a five-week externship followed by a
one-week summer institute during the second year. Participants completed the externship
at a corporate or university laboratory to develop a curriculum project under the guidance
of a field expert.
Recent scholarly work has included a focus on more intensive, content-focused
professional development programs (Scher & O’Reilly, 2009). The remaining seven studies
report shorter, intensive PD models. For example, Feuerborn et al. (2009) evaluated the
effects of a 1-week teacher PD on probability and geometry concepts. Researchers reported
that the PD led to a significant increase in teachers’ content knowledge as evidenced by the
results in pre- and post-test scores. Sixty-three teachers participated in the Zambo and
Zambo (2008) study, which examined the effects of a two-week PD workshop on the self-
50
efficacy of teacher participants. According to teacher self-reported pre- and post-surveys,
researchers noted an increase in teacher efficacy. Borasi et al. (1999) reported a study in
which general and special education teachers participated in a six-day summer institute
followed by three to four meetings during the school year. The PD program was part of a
National Science Foundation (NSF) grant project aimed at supporting secondary teachers
with mathematics reform instruction. Results of this study suggested that the PD program
was successful in supporting teachers for providing reform mathematics instruction.
The literature suggests that short, intensive PD programs along with several follow
up PD meetings can be effective for allowing for greater teacher participation as well as
promote the transfer of PD activities into classroom practice (Garet et al., 2001). Further,
PD experiences that incorporate activities for promoting collaboration enhance teacher
learning and model effective practices for teaching in inclusive classrooms (Borasi et al,
1999; Clark, 1994) Therefore, future research should include an intensive PD course, with
follow up learning experiences that promote collaboration amongst participants.
Professional Development for Special Education Teachers
Very little research specific to PD for special education mathematics teachers exists.
The two studies that were identified are reviewed here. Schumm and Vaughn (1995)
reported the findings of a series of seven PD programs during a three-year period offered
to special education teachers (elementary through high school) for the purposes of
improving their instructional practices for teaching in a general education setting.
Researchers reported several key findings including the importance of determining
teachers’ needs prior to PD implementation, focusing on specific content and effective
instructional strategies, and the benefit of collaborating with general education teachers. In
51
a qualitative case study involving four in-service special education teachers, Frey (2009)
investigated the effects of a project-based online PD experience on teacher practice and
student learning. The focus of the PD was helping teachers develop and implement
individualized academic interventions for students. The PD was offered as a 3-credit hour,
practicum experience during a 16-week semester and was primarily delivered through a
university-supported online management system. Teachers engaged in online discussion
groups and submitted regular journal entries detailing their implementation of assigned
tasks as well as student progress. Results of the study indicated that the teachers increased
their ability to provide targeted interventions and students were able to achieve academic
goals. Teachers also reported their plans to continue the practices and strategies they
learned through the PD. The gap in the literature base for providing effective PD for special
education teachers adds to the significance of the current study.
Video Cases
In addition to traditional PD courses, there has been an increase in the use of video
cases or video taped representations of teaching in teacher education programs (Kurz &
Kokic, 2012). Alsawaie and Alghazo (2010) used video lesson-analysis methodology to
measure preservice teachers’ ability to notice noteworthy classroom interactions. The
results indicated that when teachers used an online forum to discuss the video cases, they
performed better than the control group. Furthermore, the experimental group paid closer
attention to student learning and provided deeper evidence in relation to what they saw in
the video case (Alsawaie & Alghazo, 2010). Recent studies (Koc, 2011; So, Pow & Hung,
2009) have focused on video case implementation in the university classroom and their
influence on preservice teachers’ growth. The results of the study, in which preservice
52
teachers developed and analyzed video cases, indicated that video case implementation can
improve preservice teachers’ motivation, learning, empathy, and the construction of
professional identity (Koc, 2011).
Animated Representations of Teaching
Although there are various mechanisms for promoting the teacher’s ability to apply
teaching and instructional principles in the classroom, the use of classroom exemplars or
scenarios of expert teachers in practice, is a promising technique for fostering the
implementation of teacher learning to classroom practice (Moreno, 2009). These classroom
scenarios are helpful for demonstrating how knowledge about teaching and learning can be
transferred to classroom practice. According to social cognitive theory of learning
(Bandura, 1977), people tend to learn and imitate behaviors they observe, known as
observational learning. Thus, when student teachers are presented with a classroom
narrative, video, or animation modeling how an expert teacher applies principles of
teaching to his or her classroom, they will be more likely to imitate the displayed behaviors
in the future (Moreno & Ortegano-Layne, 2008). Research supports the idea of using
classroom exemplars in teacher education (Beck, King, & Marshall, 2002). Recent studies
have examined the use of video taped classroom interactions, or video cases, in teacher
education programs and inservice PD. For example, Santagata (2009) investigated the
effects of a two-year video-based PD program with secondary mathematics teachers in
low-performing schools. Throughout the program, teachers viewed and analyzed selected
portions of video taped classroom instruction. In addition, they participated in online and
face-to-face discussions and analyzed instructional lessons. Findings of this study suggest
three principles for designing video-based professional development tasks for teachers: (a)
53
attending to content-specific understanding, (b) scaffolding analysis of student thinking,
and (c) modeling a discourse of inquiry and reflection on the teaching and learning process.
In another study of 26 preservice and inservice teachers, researchers explored the
outcomes of implementing an online video-based PD program aimed at improving the
teaching of mathematics at the elementary and secondary level (Koc, Peker, & Osmanoglu,
2009). Teachers viewed and provided reflections of video cases provided through an online
forum. Findings from the study indicated that the video case discussion helped teachers
make theory to practice connections and implement what they learned in their classroom
instruction.
Although there is support for using video cases in teacher education and PD, the
research is limited for the use of animated representations of classroom experiences.
However, this tool provides a unique way to engage teachers to increase their content and
pedagogical content knowledge as well as provide a mechanism for building collaboration
amongst teacher teams. In a study of 80 student teachers enrolled in a psychology course,
participants were randomly assigned to receive one of four conditions: a) control group
(textbook narrative); b) text narrative; c) video exemplar; and d) animated exemplar.
Teachers were allowed to choose a subject (i.e., math or science) and were presented with
a classroom exemplar. Teachers were then assessed on how well they received their
respective exemplar and if the exemplar helped them to connect theory to practice. Based
on statistical results, researchers cited strong support for Bandura’s social cognitive theory
(Bandura, 1977). In addition, visual representations of classroom experiences resulted in
more positive attitudes and motivation in teachers over textual narratives. Animated
exemplars in particular were found to be a more promising tool over video as animations
54
provide greater flexibility in that they can be created to meet instructional needs with
relative ease. Moreover, the animations can de designed to address specific content and
instructional strategies while eliminating distractors such as teaching style and student
behaviors.
The proposed study uses video cases of mathematics instruction as well as allow
teachers to view and create animated representations of classroom instruction through the
Thought Experiments in Mathematics Teaching (ThEMaT) project. Several animated
experiences on geometry instruction have been created and available for viewing online,
which will serve as a basis for increasing teacher knowledge. In addition, the ThEMaT
offers an online tool called LessonSketch, which allows users to create their own animated
classroom experience. This provides a unique way to capture and demonstrate instruction
that is geared towards students with learning disabilities. As research is limited for visual
representations of teaching in inclusive classrooms, creating animations of instructional
best practices provides the opportunity to address specific learning strategies for students
and teachers. Further, teachers will be developing the animations in teams, which
promotes collaboration.
Study Implications
This literature review supports the development of the current study in several
ways. In terms of the source of the professional development, the current study offers
incentives to selected teachers. In addition, the PD provider and researcher are non-
evaluative in terms of teacher performance in the current setting so teachers feel
empowered to implement and practice what they learn.
The purpose of the professional development in the current study is to increase
55
teachers’ content and pedagogical content knowledge in congruence and similarity as well
as increasing their self-efficacy for teaching geometry in an inclusion setting. The literature
supports professional development models that seek to deepen teachers’ content and
pedagogical content knowledge, particularly when the content is aligned with district
standards and teachers’ daily work. The content covered during the PD is aligned to the
Common Core Standards, which is a statewide goal for the proposed school district. Thus,
the purpose of the PD is content specific and connected to teacher’s current practice,
elements of good practice as outlined by the literature.
The format of the current PD model is consistent with what the research indicates is
most promising in terms of promoting teacher learning. Professional development
activities that include opportunities for collective participation and collaboration, that are
linked to teachers’ daily work, and that promote active learning are more likely to be
perceived by teachers as beneficial (Birman et al., 2000; Garet et al., 2001). This PD model
addresses these organizational aspects in that general and special education teachers will
participate as co-teaching teams, develop curriculum and instructional materials that are
applicable to their current practice, and participate in engaging learning activities such as
group discussions and presentations of their work.
The literature is clear that one-time workshops or other short professional
development opportunities rarely promote improvements in teacher learning or
instruction. The proposed PD model features a 5-day summer session followed by 3 follow-
up meetings totaling approximately 30 hours of targeted learning opportunities spanning
over several months. Overall, the current study will incorporate many of the most widely
accepted features of effective professional development.
56
Chapter 3: Methodology
This chapter outlines the methodology used in this single-subject design study
examining the effects of a professional development course for increasing teacher self-
efficacy as well their content and pedagogical content knowledge in geometry. The
professional development course consisted of an in-depth exploration of congruence and
similarity concepts including the development of and proof of geometric theorems,
describing transformations, and applying trigonometric ratios to solve problems. The
course also instructed participants on implementing Universal Design for Learning (UDL)
concepts, which is a set of curriculum principles used to instruct and meet the needs of
diverse learners. The course was delivered in a blended learning format consisting of
online as well as in-person participation. A multiple probe design across three sets of two
teachers for a total of 6 participants was used in this study to demonstrate a functional
relationship between the independent and dependent variables. Qualitative data were
collected through a focus group interview at the end of the study. Additionally, social
validity data on the value of the intervention were obtained from the participants teachers.
Descriptions and justifications of the sample, design, measurement techniques,
instrumentation, and data analysis procedures are described in the sections to follow.
Participant Eligibility and Selection
Six teachers from three public charter schools in Washington, DC participated in this
study. Teachers participated as pairs and were co-teachers of a geometry course at their
respective school. To meet eligibility criteria established prior to the study, eligible
participants were: a) co-teaching partners of a secondary geometry course; and b) teaching
a classroom with at least 25% of students having an IEP. Table 2 displays demographic
57
information from the Survey on Teaching Mathematics to Students with Learning
Disabilities in Middle School, which was completed by each teacher.
Table 2 Participant Demographic Information
Gender
Male: 4 Female: 2
Number of years teaching
1-2 years: 1 3-8 years: 4 9-14 years: 1
Number of years teaching in an inclusive classroom
1-2 years: 1 3-5 years: 4 6-10 years: 1
Type of school where you teach
Urban: 0 Suburban: 0 Rural: 0 Private: 0 Public: 6
Number of students in your school
1-200: 0 201-500: 6 501-800: 0 801-1100: 0 More than 1100: 0
Average number of students in your inclusive classes:
Less than 15: 0 15-20: 2 21-25: 4 26-30: 0 31-35: 0
The number of professional development workshops related to teaching students with learning
disabilities I have been exposed to has been:
0-2: 2 3-4: 3 5-6: 1 7-9: 0 10 or more: 0
The level of administrative support for teaching an inclusive class in my school is:
Extremely Low: 0 Low: 4 Average: 2 High: 0 Extremely High: 0
The level of additional support services (e.g., counseling, resource room or teacher, instructional
materials, etc.) for teaching an inclusive class in my school is:
Extremely Low: 0 Low: 1 Average: 5 High: 0 Extremely High: 0
I used the following steps to select potential participants. First, I contacted and met
with the Principal of each school to discuss the study. During each discussion, I presented
the Principal with a letter (see Appendix A) detailing a description of the study, the
duration of the course, and any surveys and assessments each participant may be
requested to complete. If the principal agreed for me to speak to their teachers, I then
discussed the study with the general and special education geometry co-teachers and
58
presented them with a Participant Consent form (see Appendix B).
Informed Consent. Written consent was obtained from each participant using the
Participant Consent form prior to initiating the study. Each participant was informed of
their right to withdraw from the study at any time.
Human Subjects Review. Permission was obtained from the Human Subjects
Review Committee at the University of Maryland, College Park prior to beginning the
investigation.
Instructor and setting. The intervention was facilitated by the investigator. The
study took place after regular school hours inside a conference room at three different
public charter schools in Washington, DC. Each school supplied an LCD projector, and a
projection screen. The instructor also used a portable whiteboard during the intervention.
The actual intervention included four three-hour sessions after school, two six-hour
sessions on Saturdays, and two online sessions designed to take four hours each to
complete. The intervention was designed to take 32 hours to complete over a two-week
period. All instructional lessons and assessments were preplanned prior to the start of the
intervention.
Professional Development Resources
This section provides an overview of the instructional materials that were used
during the intervention to increase teacher’s self-efficacy and knowledge of geometry
teaching. Lesson plans for each session were created by the investigator using research
based evidence for providing teachers with effective PD experiences.
UDL Guidelines. The UDL guidelines (CAST, 2011) were used to provide a
framework for understanding how to create curricula that meets the needs of all learners.
59
The guidelines are designed to help teachers plan and create learning experiences that
address students’ recognition, strategic and affective learning styles. During the
intervention, the investigator presented the UDL guidelines to the participants and
demonstrated how the guidelines could be used to develop differentiated lesson plans.
Participants used the guidelines to develop lesson plans for teaching congruence and
similarity concepts. See Appendix C for a sample of the guidelines.
Videocases. During the intervention, the investigator used videocases developed by
the Learning and Teaching Geometry project (Seago et al., 2013) to engage participants in
congruence and similarity activities. The videocases offered insight into what an emerging
understanding of similarity looks like, and encouraged participants to consider specific
instructional strategies that can foster this understanding. The clips are unedited segments
selected from real classroom footage of unstaged mathematics lessons, representing a
range of grade levels, geographic locations and student populations. These clips typically
represent a conceptual hurdle or portray some degree of mathematical confusion, based on
the expectation that they are likely to provoke inquiry and discussion within the PD setting.
LessonSketch. LessonSketch is education software that helps researchers and
educators create a learning environment. Features of LessonSketch include lesson scripts,
content discussions, media authoring tools, user tracking, and assessment and course
creation tools. It is built on the idea that representations of mathematics instruction—that
is, depictions of interactions between a teacher, their students, and the content—can
anchor discussions about the teaching, learning, and mathematics, and provide grounds for
practice-based teacher development.
60
Experimental Design and Study Procedures
The current research study utilized both quantitative and qualitative methods.
Quantitative data were collected through pre- and post- measures as well as a single-
subject multiple probe design across three sets of two teachers for a total of 6 participants.
The single-subject design was used given its importance in developing evidence-based
practice in special education. Single-subject designs provide an opportunity for
researchers to test and identify an educational or behavioral intervention as evidence-
based practice (Odom et al., 2005). In the current study, a multiple probe design was
selected over a multiple baseline design because continual use of probes for teachers in
baseline would be impractical (Kazdin, 2011) given the academic nature of the
intervention. Additionally, collecting infrequent probes rather than continuous baseline
reduced the chances of participants improving their results through repeated practice
(Kazdin, 2011) and was necessary due to the short duration of the intervention. The study
consisted of 4 phases conducted in the following order: a) pre-intervention measure
administration, b) baseline phase, c) intervention phase, and d) post-intervention measure
administration.
The independent variable was introduced across each pair of teachers using a
staggered method (Horner & Odom, 2013) by collecting a baseline series of data
(consisting of at least 3 data points) for all pairs then introducing the intervention to
Stephanie and Talib only. After Stephanie and Talib reached a predetermined level of
criteria, all 3 pairs were probed (consisting of at least 3 data points). Nicole and Tiano
were then introduced to the intervention only, followed by probing all three pairs. This
pattern continued until all three groups had completed the intervention and met the
61
required criteria (Horner & Odom, 2013). This staggered approach was used to address
internal validity (Horner & Odom, 2013).
Pre- and Post-Study Instruments
The current study utilized a survey and a geometry assessment to measure teacher
self-efficacy, and content and pedagogical content knowledge in geometry both before and
after the single subject study. See Appendix D for table of measures and instruments. Each
teacher participant completed the pre- and post-measures individually in their classroom.
I reserved the classroom for several hours and contacted each teacher to arrange a time for
them to complete the measures. During the assessment, participants were given a copy of
the measure and pencil. I then explained the purpose of the measures and asked if the
participants had any questions. Participants were allowed to work on each measure
independently for as much time as needed before I collected the measure for scoring.
Teachers completed the measures in approximately 90 minutes on average.
Teacher Efficacy Measures. The Teachers’ Sense of Efficacy Scale (TSES),
developed by Tschannen-Moran and Woolfolk Hoy (2001), was used to capture teacher
self-efficacy beliefs as pre- and post-test measures. The TSES assesses the major aspects of
teacher self-efficacy through a cyclical feedback loop for efficacy judgments demonstrating
a more balanced picture of teacher self-efficacy without over generalization or deep levels
of specificity. The short form version of the TSES was selected for the current study to
decrease the total number of survey items since participants will be completing two
separate instruments to assess their efficacy beliefs in addition to completing demographic
information.
The short form consists of twelve questions focusing on three domains: self-efficacy
62
in student engagement, self-efficacy in instructional strategies, and self-efficacy in
classroom management. Each item of the survey instrument reflects pedagogical activities,
which regularly occur in an inclusive classroom. The twelve questions allow the
respondent to select the level of his or her belief along a nine-point Likert scale. Question
stems begin with the words, “How much can you do?” or “To what extent can you …”
followed by a specific pedagogical activity. Following Bandura’s (1997) nine-point
response scale, the odd numbers are labeled as follows: one is “Nothing”, three is “Very
Little”, five is “Some Influence”, seven is “Quite A Bit”, and nine is “A Great Deal”. Each point
on the scale expresses how much or how well the respondent felt he or she could do
regarding the specific task or activity (Tschannen-Moran & Woolfolk Hoy, 2001).
The short form of the TSES is scored in a manner that provides an overall self-
efficacy score for each respondent as well as an individual score on three subscales among
samples of practicing teachers. The overall self-efficacy score is computed by summing the
numeric value for the recorded responses for each of the twelve items on the self-report
instrument. The minimum number of points achievable on the TSES instrument is 12,
which represents the lowest level of efficacy a respondent can possess. The maximum
number of points achievable on the instrument is 108, which represents the highest level of
self-efficacy a respondent can possess. See Appendix E for the TSES form including a
permission letter from the developer.
Geometry Content and Pedagogical Content Knowledge Measure. The Geometry
Assessment for Secondary Teachers (GAST) assessment was used to measure changes in
teacher’s content and pedagogical content knowledge in geometry. The development of the
GAST began in October of 2008 as part of an NSF grant designed to study high school
63
geometry teacher's geometry knowledge, teaching practices, and their relationship to
student achievement. The test was designed to be predictive in nature- a high score on the
GAST assessment was meant to signify the likelihood of student success in the high school
classroom. As requested by the NSF, geometry content of the GAST assessment was
restricted to four main areas: similarity, congruence, area, and volume. The intent of the
request was to make the scope of the project less overwhelming by concentrating on these
specific areas, which are fundamental to geometry, generating results representative of
high school geometry content in general. To enhance construct validity of the assessment,
GAST team members first analyzed secondary and college geometry textbooks, state
standards, and national assessments (e.g., NAEP, ACT) to better understand the current
geometry used for teaching. Following this analysis, a team of mathematics educators,
mathematicians, high school mathematics teachers, and doctoral students developed a
blueprint for the assessment, incorporating ideas from Webb's depth of knowledge
framework. The blueprint of the assessment included three principal content areas:
Teacher Knowledge of Mathematics (30%), Teacher Knowledge of Geometric Reasoning
and Problem Solving (25%), and Teacher Knowledge of Student Learning (45%). The test
did not specifically measure all of the mathematical knowledge for teaching domains as
categorized by Ball and colleagues, but the three major sub-domains-Common Content
Knowledge, Knowledge of Content and Students, and Knowledge of Content and Teaching
were well represented. After item production, regional reviewers were assembled to
validate items for section, domain, topic, item type, and DOK level. To further ensure
construct validity for the test, both the blueprint and test items were reviewed by a panel
of national experts in geometry and mathematics teaching. See Appendix F for a sample
64
version of the GAST assessment.
Qualitative Measure
At the conclusion of the study, the six participants were invited to participate in a
focus group to discuss the outcomes of the intervention. The focus group was facilitated
and audiotaped by an independent facilitator. The researcher was not present for the focus
group. The facilitator selected was a licensed school psychologist from a nearby school
district. The focus group session lasted 30 minutes. The facilitator used an interview
questionnaire developed by the researcher to guide the session (see Appendix G).
Single Subject Data Collection
Probes. Items on the probes were created by the investigator and were directly
related to the objectives of the instructional unit. Domain probes, and objective probes
were given to participants in the study. Domain probes provided a baseline and post-
intervention assessment of participant knowledge of co-teaching principles, UDL concepts
and knowledge for teaching geometry. Objective probes provided a daily progress
monitoring of participant performance on individual objectives presented in the lessons.
Domain and objective probes were scored for the percent of accurate responses.
Domain Probes. During the baseline phase and after the intervention, participants
completed randomly chosen parallel versions of an investigator created domain probe
made up of items related to the information covered during the instructional units. Each
domain probe consisted of 10 – 15 items related to UDL principles, teaching students with
disabilities, and geometry pedagogical and content knowledge for teaching. A minimum of
two domain probes were given to participants during baseline. A minimum of three
domain probes were given at the end of the intervention phase, after a participant
65
completed all lessons and reached criterion (80%) on all objective probes in the
intervention. Three parallel versions of the domain probe were used. See Appendix H for a
sample domain probe.
Objective Probes. The objective probe included items related to the objectives
covered in a specific lesson. Objective probes were given only during the intervention
phase, at the end of the 2nd, 3rd, 5th and 6th class sessions. Each objective probe consisted of
a variety of multiple-choice questions for assessing concepts and skills covered in a specific
lesson. Objective probes totaled 8-10 points each. If a participant met criterion (80%) on
the objective probe at the end of the class session, the next lesson was presented. If
criterion was not met, the same lesson was administered the following session. See
Appendix I for a sample objective probe.
Baseline phase. The baseline condition consisted of participants completing
domain probes measuring their knowledge of geometry concepts, universal design for
learning, and pedagogical knowledge for teaching in an inclusion setting. I collected at least
three domain probes per participant during baseline. During probe sessions, participants
were given a probe, a pencil and scratch paper. I read the purpose of the probe followed by
the directions, along with asking if the participants had questions. Each participant was
allowed to work on the domain probe independently for as much time as needed before I
collected the probe for scoring. If a participant had a question about the probe items, I
responded, “Do the best you can.” When both participant’s performance reflected stability
in level and trend during baseline (i.e., at or below 60% percent criterion for at least two
consecutive data points) I began the intervention with the first pair of teachers.
Participants in the baseline condition were presented with no instruction during that
66
period.
Intervention. The independent variable in a single case design study must be
actively manipulated to document experimental control (Horner et al., 2005). In the
current study, the independent variable (i.e. the professional development course),
included features of professional development cited in the literature base and incorporated
UDL practices. The professional development course consisted of an in-depth exploration
of congruence and similarity concepts including the development of and proof of geometric
theorems, describing transformations, and applying trigonometric ratios to solve problems.
The course was comprised of online as well as in-person participation. A course outline is
included in Appendix J.
The in-person portion of the course consisted of three lessons covered over a three
day period and instructed participants on best practices for co-teaching in inclusive
settings as well as featured instruction on two modules created through the Learning and
Teaching Geometry Project (Seago et al., 2013). Lesson 1 of the professional development
course provided instruction on co-teaching and inclusive practices focused on effective co-
teaching models, and incorporating UDL principles into teacher lesson plans. Participants
viewed video examples, participated in role-play, and modified actual lesson plans. Lesson
2 examined the meaning of defining congruence and similarity through transformations as
articulated in the Common Core State Standards. I guided teachers through a constructed
problem similar to what they could present to their students. Teachers then discussed and
developed strategies for how they would teach it in their classroom. Lesson 3 investigated
the connection between similarity, slope, and the graphs of linear functions through
mathematical tasks, analyzing student thinking, and exploring a computer-based applet.
67
Each of the two geometry instructional modules presented in Lessons 2 and 3 contained a
sequence of videocases in which specific and increasingly complex mathematical ideas
were presented within the dynamics of classroom practice. The videocases offered insight
into what an emerging understanding of similarity looks like, and encouraged teachers to
consider specific instructional strategies that can foster this understanding. Appendix K
contains the lesson plans for the in-person sessions.
The online section of the course was facilitated through the use of the LessonSketch
tool provided by the Thought Experiments in Mathematics Teaching (ThEMaT) project.
LessonSketch is a website for practice-based professional development of secondary
mathematics teachers as well as a forum for ongoing conversations about mathematics
instruction. It supports the creation, examination, and discussion of scenarios that depict
the practice of mathematics instruction. Teacher participants viewed and discussed
animated sketches about geometry concepts as well as created lesson plans using the
online tool. The online portion of the course consisted of two days. Each day, participants
viewed an animated representation of geometry instruction and responded to a series of
questions through an online discussion board or animated script. Figure 1 displays two
screenshot examples of an animation from LessonSketch. Using the animated scripts,
participants provided feedback and commented on the animations as well as offered
suggestions for specific teacher dialogue that would enhance instruction. For example, in
response to a geometry animation on congruent triangles, Stephanie suggested that the
teacher help student tend to precise measurement by matching the vertices of two
congruent triangles.
68
Figure 1 Animated Script Example
Inter-assessor reliability. Inter-assessor reliability was obtained for 100 percent
of the data points from the domain probes. A trained graduate student and I independently
scored each probe. The percentage of scorer agreement was calculated by dividing the
number of agreements of correct by the number of disagreements and agreements and
multiplying the result by 100 percent (Kennedy, 2005). Reliability was 100 percent across
all domain probes for each teacher.
Fidelity of treatment. Two independent observers reviewed the video recorded PD
lessons for treatment fidelity using a checklist (Appendix L) containing critical features of
the intervention (O’Donnell, 2008). I trained the independent observers by explaining the
checklists and reviewing the written lesson plans, assessments, and recorded sessions. I
conducted two mock instructional sessions where I followed my lesson script and the two
observers were instructed to note either a “1” (if the component was present) or a “0” (if
the component was not present) for each component on the fidelity checklist during the
session. I compared the two checklists and reviewed any discrepancies. Observers
69
maintained 100% agreement over the mock sessions.
Fidelity observations were conducted on 33% of the instructional and assessment
sessions using video recordings of the sessions by two independent observers (Kennedy,
2005). The fidelity of treatment for each session was calculated by dividing the number of
observed components by the number of total components on the checklist and multiplying
the result by 100 (O’Neill, et al., 2011). The fidelity of treatment for the study was obtained
by adding the percentages for each session and dividing the sum by the number of
observations. Treatment fidelity was 100 percent.
Inter-assessor agreement on the objective probes was obtained by collecting the
scores of the two assessors and calculating the percentage obtained by dividing the smaller
score by the larger score and multiplying by 100, yielding a total percent agreement (Gast,
2010). Interobserver agreement for the study was 100 percent.
Data analysis procedures
In single-subject research, experimental control is established when visual analysis
of the data demonstrates a functional relationship between independent and dependent
variables. A functional relationship must be established where systematic manipulation of
an independent variable has a consistent effect on a dependent variable (Kennedy, 2005).
In the present study, a functional relationship was determined based on the improvement
of participants’ content and pedagogical content knowledge for teaching geometry
concepts, and whether the change was caused by the treatment. Visual analysis of graphed
data was used to determine the: (a) stability of the baseline conditions, (b) rapidity of
changes in the variables between conditions (baseline and post-intervention), (c) changes
in the mean performance and changes in pattern of individual data points between
70
conditions, and (d) variability in the level and/or trend within and across conditions
(Kennedy, 2005).
Statistical tests selected to analyze the collected data are frequency counts and non-
parametric tests. Data analysis for the three research questions was conducted using the
data sources outlined in Table 3.
Table 3 Data Sources and Analyses
Research Question Data Sources Analysis
What is the effect of a teacher
professional development course
involving use of animated
representations addressing
congruence and similarity on special
education and mathematics
education teacher self-efficacy,
content knowledge, and pedagogical
content knowledge in an inclusion
setting?
Geometry Assessments for Secondary Teachers (GAST) Domain and Objective Probes; Teachers Sense of Efficacy Scale (TSES)
Single-subject analysis Non-parametric tests Frequency counts
How do general education and
special education teachers compare
in their understanding of congruence
and similarity as well as their self-
efficacy for teaching geometry in an
inclusive setting?
Geometry Assessments for Secondary Teachers (GAST); Teachers Sense of Efficacy Scale (TSES)
Non-parametric tests
To what extent do general and
special education teachers find a
blended learning professional
development course as an effective
means for increasing teacher content
and pedagogical content knowledge
in geometry?
Social Validity Measure Focus Group
Mean scores Focus Group Notes
Social validity. Social validity refers to the degree of social acceptance of the importance,
effectiveness, and appropriateness of an intervention (Carter, 2010; Foster & Mash, 1999).
71
Horner and colleagues (2005) suggested that the social validity of research goals can be
enhanced through design features that include: a) dependent variables that have high
social importance, b) independent variables that can be applied with fidelity by typical
intervention agents, c) procedures that are acceptable, feasible, and effective as reported
by typical intervention agents, and d) an intervention that meets a defined need. At the
conclusion of the study, participants completed an investigator-developed survey
(Appendix M) based on other social validity measures (Mulcahy, 2007; Strickland, 2011)
which assessed general likes and dislikes about the intervention along with perceptions of
specific design features (i.e. video cases and online discussions).
72
Chapter 4: Results
The purpose of this study was to investigate the effects of a professional development
course for increasing teacher self-efficacy as well their content and pedagogical content
knowledge in geometry. This chapter presents the profile of participants used for the
research and a summary of the findings relative to each of the research questions posed in
Chapter 1. Finally, I report the results from social validity measures across participants.
The research questions included the following:
1. What is the effect of a teacher professional development course involving use of
animated representations addressing congruence and similarity on special
education and mathematics education teacher self-efficacy, content knowledge, and
pedagogical content knowledge in an inclusion setting?
2. How do general education and special education teachers compare in their
understanding of congruence and similarity as well as their self-efficacy for teaching
geometry in an inclusive setting?
3. To what extent do general and special education teachers find a blended learning
professional development course as an effective means for increasing teacher
content and pedagogical content knowledge in geometry?
Participant Profile
Participant demographic data were collected using Part I of the Survey on Teaching
Mathematics to Students with Disabilities in Middle School Survey, which was summarized
in Chapter 3. Teacher participants were co-teachers in their respective schools and
participated in the intervention in pairs. Stephanie is a special education teacher and has
73
taught with Talib, a general education mathematics teacher, for two years. Nicole is a first
year special education teacher and this is her first year working with Tiano, a mathematics
teacher. Reggie is a special education teacher and this is his first year working with
Gregory, a general education mathematics teacher.
Results on Professional Development Course
In this section, I report results as they relate to each research question. First, I
provide the research question, and then I present each participant’s results, including mean
percent accuracy increases from baseline to post-intervention, and the range of scores for
each participant. Results are presented graphically for each participant for percent
accuracy on domain probes, which were administered during baseline and post-
intervention phases, as well as percent accuracy on daily objective probes, which were
administered during the intervention phase.
Research Question 1
What is the effect of a teacher professional development course involving use of animated
representations addressing congruence and similarity on special education and mathematics
education teacher self-efficacy, content knowledge, and pedagogical content knowledge in an
inclusion setting?
As noted in Chapter 3, increases in teacher efficacy, content knowledge and
pedagogical content knowledge in an inclusion setting were measured by pre-post surveys
as well as participant performance on domain and objective probes.
Pre-/Post- Survey
The TSES was used in order to discover if any significant changes occurred in
teacher self-efficacy for participants from pre-test to post-test. For the TSES, survey items
74
are factored by one of three categories: Efficacy in Student Engagement, Efficacy in
Instructional Practices, and Efficacy in Classroom Management. To determine the subscale
score for each factor, the unweighted means that load on each factor are computed. Table 4
shows the pretest and posttest unweighted means for each participant in each factor.
Table 4 TSES Results
Efficacy in Student
Engagement
Efficacy in Instructional
Practices
Efficacy in Classroom
Management
Pre-Mean Score
Post
Pre
Post
Pre
Post
Stephanie 8 8.25 8 8.75 8.5 8.5
Talib 7.5 8 6.75 8 7.75 7.75
Nicole 6.5 6.75 6 7.75 6.75 7
Tiano 7 7.5 6.75 7.75 7.75 8
Reggie 7.5 7.5 7.75 8.5 6.75 6.75
Gregory 6.25 6.5 6 7.75 7 7.25
Domain and Objective Probes
All participants improved their mean percent accuracy from baseline to post-
intervention on the domain probes, and each participant met criterion of 80%. Figure 1
displays the percent accuracy on domain and objective probes during baseline,
intervention, and post-intervention phases for each pair of participants. Visual analysis of
each graph reveals marked changes in level from baseline to post-intervention for all
participants. Further, there is low variability in data points in both the baseline and post-
intervention phases, reflecting stable data for each participant. Stephanie increased 76
percentage points in mean percent accuracy, Talib increased 68 percentage points, Nicole
increased 83 percentage points, Tiano increased 80 percentage points, Reggie increased 77
percentage points, and Gregory increased 77 percentage points.
75
Figure 1 Percentage of Accurate Responses on Domain Probes
Baseline Intervention
Post Intervention
Session
Session
Session
0Stephanie
υ Talib
σ Nicole
ν Tiano
λ Reggie
ν Gregory
76
Stephanie. During baseline, Stephanie earned a mean score of 17% on domain
probes, (r= 10% - 20%). Following the intervention, she earned a mean score of 92% (r =
85% - 100%).
Talib. During baseline, Talib earned a mean score of 27% (r = 20% - 30%).
Following the intervention, he earned a mean score of 94% (r = 90% - 95%).
Nicole. During baseline, Nicole earned a mean score of 7.5% on domain probes (r =
5% - 10%). Following the intervention, she earned a mean score of 90% (r = 85% - 95%).
Tiano. During baseline, Tiano earned a mean score of 16% (r =10% - 20%).
Following the intervention, he earned a mean score of 96% (r = 90% - 100%).
Reggie. During baseline, Reggie earned a mean score of 8% (r =5% - 10%).
Following the intervention, he earned a mean score of 85% (r = 80% - 90%).
Gregory. During baseline, Gregory earned a mean score of 16% (r =10% - 20%).
Following the intervention, he earned a mean score of 93% (r = 90% - 100%).
Daily objective probes were administered throughout the intervention phase.
Figure 2 displays the results of the objective probes for each participant pair. Visual
analysis reveals stability across all data points for all participants. In addition, the graphs
demonstrate high levels of performance across all participants.
77
Figure 2 Percentage of Accurate Responses on Objective Probes
70
75
80
85
90
95
100
1 2 3 4
Pe
rce
nta
ge
Ac
cu
rate
Re
spo
nse
s
Session
Stephanie
Talib
70
75
80
85
90
95
100
1 2 3 4
Pe
rce
nta
ge
Ac
cu
rate
Re
spo
nse
s
Session
Nicole
Tiano
74
76
78
80
82
84
86
88
90
92
1 2 3 4
Pe
rce
nta
ge
Ac
cu
rate
Re
spo
nse
s
Session
Reggie
Gregory
78
Research Question 2
How do general education and special education teachers compare in their understanding
and knowledge of congruence and similarity as well as their self-efficacy for teaching
geometry in an inclusive setting?
Participant understanding and knowledge of congruence and similarity concepts
was measured using the GAST assessment. The assessment consists of 15 multiple choice
and 11 constructed response items. The maximum score obtained on the assessment is 48
points. The assessment was scored by an independent researcher who was trained by the
assessment developer. Table 5 reports the pre and post mean scores within groups for
special education teachers and general education teachers based on their responses on the
GAST assessment.
Table 5 – Within Group Pre and Post Mean Scores on GAST Assessment of
Congruence and Similarity
Pre-Test Score Post-Test Score Change
Special Education Teachers
Stephanie 20 29 +9
Nicole 16 22 +6
Reggie 13 15 +2
Special Education Group Mean 16.33 22 +6.33
General Education Teachers
Talib 32 35 +3
Tiano 36 37 +1
Greg 27 28 +1
General Education Group Mean 31.67 33.33 +1.66
79
To examine how special education and general education teachers compare in their
self-efficacy, the average subscale score for each dimension on the TSES were calculated for
special education and general education teachers separately (see Table 6). Based on
participant responses, special education teachers generally had higher self-efficacy scores
compared to their general education counterparts in two out of the three dimensions.
Table 6 Average Pre and Post TSES Scores By Dimension Special Education Teachers General Education Teachers
Efficacy for Student Engagement 7.4 7.1
Efficacy for Instructional Practices 7.8 7.4
Efficacy for Classroom Management 7.4 7.6
Research Question 3
To what extent do general and special education teachers find a blended learning professional
development course as an effective means for increasing teacher content and pedagogical
content knowledge in geometry?
Based on focus group notes participants felt that there were three aspects from this
professional development experience that were different than other professional
development experiences they shared. They were a) the opportunity to work through the
video cases, b) the use of research to support the content in the course, and c) the exposure
to the LessonSketch tool.
Participants felt that the use of video cases in the course enabled them to use these
same experiences in their classroom. Participants said that while taking the class they were
often placed in the role as the student. “The instructor often wanted us to see how this
strategy works through the eyes of a student,” one teacher remarked. Another teacher
commented, “the videos provided examples that could be brought right back to the
classroom”.
80
The use of research supporting the UDL content was also important to the
participants. In several responses participants said that the use of the UDL guidelines
validated the strategies that was being discussed. One teacher commented, “people think
often that we just pick up strategies and they are not research-based and I think the
research that has been shared in this course proves that it can work.”
Participants also found the LessonSketch tool to be a “creative way to write lessons
and learn new concepts”. The LessonSketch tool allowed teachers to work together to
create a lesson script as well as view online animated representations of geometry
instruction. One teacher commented, “I think my students would enjoy using this tool to
design math problems in class”.
Social Validity
The average scores on the social validity measure ranged from 4 to 5 (see Table 7).
Overall, participants responded that they strongly agreed that the professional
development course relevant and useful and that the activities were engaging. While the
special education teachers were somewhat neutral that the course increased their
understanding and use of Universal for Design for Learning strategies (M=3.33), general
education teachers strongly agreed that the intervention increased their knowledge and
understanding of UDL concepts (M=5).
81
Table 7 – Social Validity Measure Results
In-Person Sessions
Stephanie Talib Nicole Tiano Reggie Gregory Special
Ed
Teacher
Mean
General
Ed
Teacher
Mean
The professional development content was relevant and useful
5 5 5 5 5 5 5 5
The activities were engaging and connected to the professional development objectives
4 5 5 5 5 4 4.67 4.67
I gained knowledge about Universal Design for Learning concepts
3 5 4 5 3 5 3.33 5
I gained knowledge about using Universal Design for Learning concepts in my lessons
4 5 4 5 3 5 3.67 5
I gained knowledge about similarity and congruency concepts in geometry
5 4 5 3 5 4 5 3.67
I will be able to apply the strategies discussed in this professional development immediately in my teaching
5 5 4 5 4 4 4.33 4.67
The video case studies were effective for learning about teaching geometry concepts
5 5 5 4 5 5 5 4.67
Online Sessions
Stephanie
Talib
Nicole
Tiano
Reggie
Gregory
Special
Ed
Teacher
Mean
General
Ed
Teacher
Mean
The animated representations of teaching were effective for facilitating pedagogical discussions
4 5 5 5 5 4 4.67 4.67
The online activities were effective for facilitating collaboration
4 4 5 5 4 5 4.33 4.67
Participants also responded favorably to the intervention in open-ended questions.
For example, Nicole stated that the best part of the course was “completing a variety of
activities.” Gregory reported that “learning to incorporate UDL strategies” was the best part
of the course. When comparing the in-person and online sessions, Tiano reported that the
animated representations “was a unique way to learn a concept.”
82
Chapter 5: Discussion
The purpose of this study was to examine the effects of a co-teacher professional
development course for increasing the knowledge and self-efficacy of special education and
general education geometry teachers in an inclusion setting. The participants learned best
practices for co-teaching, how to apply UDL strategies and concepts, and gained knowledge
for teaching congruence and similarity concepts.
This chapter summarizes the major findings of the study, presents the study
limitations, provides recommendations that result from this study, and offers opportunities
for future research.
Study Design and Rationale
A multiple probe design across three sets of two teachers for a total of 6 participants
was used in this study to demonstrate a functional relationship between the independent
and dependent variables. In addition, the study utilized several instruments to gather
teacher demographic data, and measure teacher self-efficacy, content and pedagogical
content knowledge in geometry both before and after the single subject study.
The major themes that guided this study were the following: the effectiveness of
various professional development delivery systems; the importance of teachers’
perceptions of their effectiveness in the classroom; and the mathematical knowledge
teachers need to effectively teach congruence and similarity concepts.
The literature concerning professional development for educators from
Garet et al (2001), the National Council for Teachers of Mathematics, the National Research
Council , and the Common Core Standards for Mathematics provided the frameworks for
the researcher’s design in the professional development delivery activities used in the
83
study. Changing teachers’ perceptions of mathematics instruction and keeping teachers
current in mathematics practices has been a major issue in education in the United States
(TIMSS, 2007). In addition, teachers are instructing students with diverse learning needs.
In order for teachers to meet the diverse learning needs of students, teachers must believe
in their own ability to do so. Self-efficacy, in regards to education, plays a critical role in the
instructional decisions teachers make on a consistent basis (Kosko & Wilkins, 2009).
The content delivered in the professional development treatment was based on the
Common Core State Standards Mathematical Practices for Teaching, and the work of Ball et
al (2008) on investigating what mathematical knowledge is needed for effective teaching,
and the research base for best practices for teaching in an inclusive setting. The testing
instruments used in this study consisted of a pre and post assessment two surveys as well
as researcher developed probes for single-subject analysis. The GAST assessment was used
to measure participant knowledge and understanding of geometry concepts before and
after the intervention. The first survey, the Teachers’ Sense of Efficacy Scale (Tschannen-
Moran and Woolfolk Hoy 2001), was used to capture teacher self-efficacy beliefs as pre-
and post-test measures in three dimensions; student engagement, instructional strategies,
and classroom management. The Survey on Teaching Mathematics to Students with
Learning Disabilities in Middle School (DeSimone & Parmar, 2006) is a second survey used
measure teacher beliefs related to teaching mathematics to students in inclusive settings.
The professional development course consisted of an exploration of congruence and
similarity concepts including the development of and proof of geometric theorems,
describing transformations, and applying trigonometric ratios to solve problems. In
addition, participants received instruction on the best practices for teaching in an inclusive
84
setting. The course was delivered in a blended learning format, consisting of online as well
as in-person participation.
Interpretation of Findings
Research question one: What is the effect of a teacher professional development
course involving use of animated representations addressing congruence and similarity on
special education and mathematics education teacher self-efficacy, content knowledge, and
pedagogical content knowledge in an inclusion setting? In regards to teacher content and
pedagogical content knowledge, the success of this intervention is demonstrated in the
results each participant experienced from baseline to post-intervention phase on the
domain probes. Stephanie increased 75 percentage points from baseline to post-
intervention. Talib increased 67 percentage points from baseline to post-intervention.
Nicole, who is a first year special education teacher, increased 82.5 percentage points from
baseline to post-intervention. Tiano, an experienced mathematics teacher, increased 80
percentage points from baseline to post-intervention. Reggie increased 77 percentage
points from baseline to post-intervention. Gregory, who has been teaching mathematics for
two years, increased 77 percentage points from baseline to post-intervention.
The ability of each of the participants to perform at such high levels both during
instruction and during the post-intervention domain probe sessions suggests that the
professional development course, presented through blended learning activities, positively
affected teacher content and pedagogical content knowledge for teaching geometry
concepts in an inclusive setting. Furthermore, the consistent changes in outcomes for
Stephanie and Talib demonstrated the independent variable had a positive impact on the
dependent variable, which establishes a functional relationship (Kennedy, 2005).
85
Replication of these findings with the remaining teacher pairs demonstrated robustness of
the experimental control and generality to other participants (Kennedy, 2005).
Further, the post-test mean scores on the GAST assessment, which measures
teacher content and pedagogical content knowledge in geometry, increased for the
participants by 13 points. This supports the evidence that the professional development
course was effective for increasing teacher content and pedagogical content knowledge.
In regards to teacher self-efficacy, post survey results on the TSES demonstrate an
increase in each of the three dimensions: Efficacy for Student Engagement, Efficacy for
Instructional Practices, and Efficacy for Classroom Management. However, the increase in
Efficacy for Instructional Practices was more measurable compared to the other two
dimensions. This suggests that while the professional development course was beneficial
for improving instructional practices, the course was not as effective for improving
participants’ ability to improve their classroom management skills or increase student
engagement.
Research question two: How do general education and special education teachers
compare in their understanding and knowledge of congruence and similarity as well as
their self-efficacy for teaching geometry in an inclusive setting? In regards to teacher
knowledge of congruence and similarity concepts, general education teachers
outperformed special education teachers, based on GAST assessment results. This
supports the literature, which suggests that special education teachers generally lack
content specific knowledge. It should be noted that while Tiano, a mathematics teacher
obtained the highest score on the GAST assessment (37 out of 48), this score was less than
80% proficiency. In addition, the special education teachers demonstrated higher
86
increases from pre to post-test compared to general education teachers. Yet, the average
score for special education teachers on the GAST post assessment was less than 50%
proficiency. Special education and general education teachers alike reported that they felt
that the GAST assessment was particularly challenging.
With regard to teacher self-efficacy, special education teachers reported higher self-
efficacy in student engagement and instructional practices. This suggests that special
education teachers feel more confident in their ability to motivate students as well as
provide a variety of instructional strategies. As special education teachers receive specific
training in these areas, it makes sense that these teachers would report a higher self-
efficacy.
Research question three: To what extent do general and special education
teachers find a blended learning professional development course as an effective means for
increasing teacher content and pedagogical content knowledge in geometry? Based on
focus group notes participants noted several positive aspects of the professional
development including the use of video cases, the application of research-based strategies
and the use of the LessonSketch tool. In addition, teacher participants felt that the
collaborative piece that allowed teacher pairs to work together was the major support in
making their professional development experience a positive one. The fact that the
participants were already co-teachers and participating with their partner may have
enhanced the collaborative nature of the course. During the course, I observed Talib and
Stephanie’s willingness to work through the activities together with less hesitation than the
other co-teaching partners. This is indicative of their current working relationship. The
other co-teaching partners had only been together for the current school year. In addition,
87
Reggie, the most experienced teacher, was more reticent in his approach to co-teaching,
often allowing Gregory to take the lead on collaborative activities in the course. This was
an indication of the type of role Reggie perceives of himself as a special education co-
teacher.
The participants also felt that having a common experience as participating in the
same professional development allowed conversations to be geared toward their new
learning. The use of the LessonSketch helped to facilitate collaboration as teachers worked
together to discuss the animated representations.
Summary
This study suggests that a co-learner blended learning professional development
course can improve teacher content and pedagogical knowledge in geometry as well as
increase teacher efficacy for teaching in an inclusive setting. Each participant dramatically
improved his or her performance on domain probes from baseline to post-intervention. In
addition, participants improved their scores on post assessments as well as reported
favorable outcomes in the focus group interview and social validity measure.
Study Limitations
There are several limitations to this study. One limitation was the length of the
professional development course. The average duration of the intervention was 30 hours.
Although the literature suggests that short, intensive PD programs can be effective for
allowing for greater teacher participation as well as promote the transfer of PD activities
into classroom practice (Garet et al., 2001), the short duration of the current study may
warrant the need for additional follow up sessions. Another limitation of the study is that
student achievement was not measured as a dependent variable. The research is limited
88
for studies that link student performance to professional development outcomes. Further,
this study did not include observations of teachers in their normal setting after completing
the professional development course. Follow-up observations would help determine
teacher application of the strategies they learned during the course.
Given that the investigator implemented the intervention, my own biases may have
influenced the study. I attempted to control for those biases through the use of scripted lessons,
explicit instructional procedures, and fidelity of treatment procedures. It is also noted that while
participant scores increased significantly on domain and objective probes, there was less of an
increase in scores on the GAST post-test. This may indicate that the probes were heavily aligned
to the intervention.
Directions for Future Research
This study could advance the body of knowledge on the effects of a co-teacher
professional development model in relation to teachers’ specialized mathematics
knowledge for teaching. While the current study’s design was intended to examine how a
blended format professional development delivery model could affect teachers’ self-
efficacy and knowledge in geometry. The co-teacher model was found to allow
collaborative inquiry to occur; a major component in the institutionalization of any practice
(Guskey, 2000). This study could have the potential to direct future research on educating
teacher partners to build collaboration and promote sustainability in their current setting.
The targeted population for this study was current teachers. However, it is
suggested that future research include pre-service teachers as part of their teacher
preparation program. Including special education and general education pre-service
teachers in the same course may enhance collaboration and increase content and
89
pedagogical knowledge for teachers. For example, a study could be conducted to find if
there is a significant difference in influencing preservice teachers’ self-efficacy of
mathematics instruction between taking courses with peers and conducting traditional
field work requirements, or taking the course and conducting fieldwork requirements with
a co-teacher candidate. While the current study focused on mathematics content, future
research could find the feasibility of using the co-learner model in teaching other
disciplines.
Although the duration of study may have limited the intervention’s influence on
student achievement, it may have been beneficial to include student performance data for
the teachers before and after the intervention. The single-subject design of the study is a
limitation as most professional development courses involve groups of teachers. In the
current student study, there were limited opportunities for distractions by other individuals, and
since the room contained only the materials and equipment used for the intervention, there were
no other distracting items in the room. Additionally, the teachers had the sole attention of the
instructor, which is atypical of a professional development course. Under these controlled
conditions, the participants demonstrated success. However, replication in a larger setting is
necessary to generalize the results to common professional development courses.
Opportunities to determine any effects of teacher professional development on student
achievement are needed as well. As noted in the literature review, there is limited research
linking professional development to student achievement. This particular study could be
extended to include student performance scores for those teachers participating in the
professional development course.
90
Conclusion
The self-efficacy beliefs of teachers play a significant role in teacher’s ability to adapt
their instruction to meet the needs of a diverse student population. In order to ensure that
all students improve their conceptual knowledge of geometry content, teachers must have
a deep understanding of geometry content knowledge as well as strong pedagogical
content knowledge. The current study investigated the effects of a blended professional
development course for increasing teacher self-efficacy as well their content and
pedagogical content knowledge in geometry. The results of this study provide initial evidence
that a short, intensive professional development course can be effective for improving teacher
self-efficacy and knowledge in geometry concepts. The co-teacher participant design of the
course also promoted collaborative inquiry, which is a major component for effective
professional development outcomes (Guskey, 2000). Continued research is critical to identify
opportunities to provide additional courses that involve general and special education co-teachers
learning together in other disciplines.
91
Table 1 Summary of Studies
Authors
(Date)
Purpose Sample/Participants Research Design
and Intervention
(IV)
Dependent
Variable
Results
Arbaugh (2003)
Increase content knowledge and pedagogical content knowledge in geometry
7 high school mathematics teachers
Qualitative Design Study groups 10 Bi-weekly study groups sessions
Individual interviews, audio tape accounts of study group sessions
Teachers reported increase self-efficacy and confidence in teaching geometry concepts
Borasi, Fonzi, Smith & Rose, (1999)
Increase content knowledge and pedagogical content knowledge in geometry
39 general and special education teachers in mathematics
Qualitative Design 6-day PD summer institute followed by at least two follow up meetings during the school year.
Qualitative measures (surveys, journals, interviews, transcriptions of audio recordings
Teachers reported increase in use of inquiry-based instruction. Heterogeneous groups of general and special education teachers was a benefit.
Brown & Benken (2009)
Increase content knowledge and pedagogical content knowledge in geometry
3 high school mathematics teachers
Qualitative Design Two integrated courses on content and pedagogy. Courses met weekly
Transcripts of instructional sessions and informal meetings, field notes and participant course assignments
Pre-tests showed teachers had gaps in content knowledge. Focus on content and collaboration is needed in professional development.
92
Feuerborn, Chinn, & Morlan, 2009;
Increase content knowledge and pedagogical content knowledge in geometry
55 middle school mathematics teachers
Mixed Study Two 1-week professional development courses given over the summer. Course one on algebraic sense. Course two geometric concepts, probability and statistics.
Pre/post content test. Self-report survey data involving teacher perceptions, reactions, and reflections.
Post test results showed significant increase in teacher content knowledge. Teachers reported minimal change in self-efficacy.
Kimmel & Deek (1999)
Increase teacher ability to meet the needs of students with disabilities
84 secondary teachers (3 cohorts of 23 each)
Comprehensive professional development model delivered during the summer and academic year.
Teacher self-reported data
The program led to development of coping skills and persistence in the teaching of science and math for all students.
Merrill, Devine, & Brown, 2010
Increase content knowledge and pedagogical content knowledge in geometry
8 mathematics teachers
More than 80 hours of professional development on using 3-D solid modeling software for instructing geometry concepts.
Teacher self-reported evaluation data
Teacher participants reported the value of the PD experience 4.5/5.0. Teachers also reported that the PD improved their instructional practice.
93
Powell-Moman & Brown-Schild (2011)
Increase teacher self-efficacy
23 mathematics teachers
Two-year STEM professional development program taught by science expert.
Demographics; self-efficacy pre/post survey
Post test results show significant improvement in teacher self-efficacy
Ross & Bruce (2007)
Increase teacher self-efficacy
106 grade 6 mathematics teachers
Professional development consisted of 1 full day followed by three 2-hr after school sessions. Focus was on communicating mathematical ideas.
Teacher Sense of Efficacy Scale
Treatment group outperformed control group in 3 out 4 measures of self-efficacy. Results were statistically significant for only teacher efficacy in classroom management
Swackhamer, Koellner, Basile, & Kimbrough, 2009
Increase teacher self-efficacy
88 secondary mathematics teachers
Teachers completed at least one of 15 content based math and science courses designed to increase content and pedagogical content knowledge.
Science Teaching Efficacy Belief Instrument
Survey results report that teacher’s efficacy for teaching was higher for those teachers who attended at least 4 courses.
Zambo & Zambo, 2008
Increase teacher self-efficacy
63 4th through 10th grade mathematics teachers
Two-week summer professional development workshop on mathematics problem solving.
Collective Efficacy Questionnaire (Goddard, Hoy, & Woolfolk-Hoy, 2000); Enoch & Riggs
Post survey results report increase in teacher personal competence compared to collective
94
Elementary Science Efficacy Questionnaire (1990)
competence (belief in their colleagues)
95
Appendix A: Principal Letter
October 1, 2014
Dear School Leader,
I am writing to request permission to provide targeted professional development to several of your teachers as
part of a research project I completing under my doctorate program at the University of Maryland College Park.
My advisor is Dr. Peter Leone and the purpose of this research project is to study the effects of a professional
development course for secondary geometry teachers and special education teachers for the purpose of
increasing teacher self efficacy as well as their content and pedagogical content knowledge in geometry.
The professional development course will take place during the 2014 Fall semester. The course will be
delivered in a blended format consisting of both in-person and online participation. The total duration of the
course is 32 hours over a two-week period. Teachers will be provided with learning materials and resources
needed to participate in the course. Lesson topics will focus on the teaching of similarity and congruence in
geometry. During the intervention, teachers will be asked to complete several brief assessments to measure
their understanding of the material presented. Each assessment should take no more than 15 minutes to
complete. Three 90-minute follow-up sessions will be conducted for intervention maintenance during the
semester.
At the beginning and conclusion of the study, teachers will be asked to complete an assessment to measure their
understanding of Geometry concepts as well as complete two surveys. The surveys will collect specific
demographic data about their teaching experience as well as their beliefs about their personal teaching
effectiveness. Altogether, the assessment and surveys should take no more than 90 minutes to complete. I have
enclosed a Teacher Participation Consent form for more detailed information.
I hope you will allow me to offer this professional development opportunity to your teachers. Please let me
know if you have any questions and I look forward to meeting with you to further discuss my research project.
Sincerely,
Kenneth Wright
Principal Research Investigator
96
Appendix B: Participant Consent Form
Project Title
Teaching for Inclusion: The Effects of a Professional Development Course for Secondary General and
Special Education Mathematics Teachers for Increasing Teacher Knowledge and Self-Efficacy in Geometry
Purpose of the Study
This is a research project being conducted by Mr. Kenneth Wright, Vice-Principal at Meridian Public
Charter School, as part of his doctoral studies at the University of Maryland: College Park, under the
supervision of Dr. Peter Leone. The purpose of this research project is to study the effects of a
professional development course for secondary geometry teachers and special education teachers for
the purpose of increasing teacher self efficacy as well as their content and pedagogical content
knowledge in geometry. You are being asked to participate to serve as a typical intervention agent, or
someone who would possibly receive this professional development as a non-researcher.
What will I be asked to do?
At the beginning and conclusion of the study, you will be asked to complete an assessment to measure
your understanding of Geometry concepts as well as complete two surveys. The surveys will collect
specific demographic data about your teaching experience and your beliefs about your teaching
effectiveness. Altogether, the assessment and surveys should take no more than 90 minutes to complete.
During the intervention, you will be asked to complete several brief assessments to measure your
understanding of the material presented. These assessments should take no more than 15 minutes to
complete.
The course will be delivered in a blended format consisting of both in-person and online participation.
The total duration of the course is 32 hours over a two-week period. You will be provided with learning
materials and resources needed to participate in the course. Lesson topics will focus on the teaching of
similarity and congruence in geometry. Three 90 minute follow-up sessions will be conducted for
intervention maintenance during the semester.
Additionally you will be asked your opinion of the intervention. For example you will be asked if you
think the intervention helped you as a teacher to provide better instruction to your students and what
you liked most and least about the intervention. Participants will receive $100 as reimbursement for your
time and travel expenses upon conclusion of the study.
During the study, we will be video recording the instructional sessions. We will use portions of this video
to 1) determine if the intervention is being implemented as planned; and 2) provide information for
research presentations, publications, and/or teacher trainings.
Potential Risks
Risks include possible frustration with some of the assessments and tasks and the possibility of teachers’ image being viewed in future research presentations and/or teacher trainings. In addition, while I am a Vice Principal for Meridian Public Charter School, I will have no influence on participant employment or evaluations.
Potential Benefits
Potential benefits include acquiring new knowledge and strategies for improving instructional practices
97
related to teaching similarity and congruence concepts.
Confidentiality
All information collected by this study will be kept confidential to the extent permitted by law. All data collected will be kept in a locked file cabinet in my office at Meridian Public Charter School or digitally on a password protected hard drive. If we write a report or article about this research project, your identity will be protected to the maximum extent possible and your name will not be used. Data will be identified using false names or an identification code. Your information may be shared with representatives of the University of Maryland: College Park, or governmental authorities if we are required to do so by law.
Do I have be in this research? Can I stop participating at any time?
Your participation in this research is completely voluntary. You may choose not to take part at all. If you
decide not participate in this study you will not be penalized. If you decide to participate in this research,
you may request to stop participating at any time. If you withdraw from the study prior to the conclusion,
you will still be reimbursed the same amount/your reimbursement will be pro-rated based on the
number of days you participated in the study.
What if I have Questions?
This is a research project being conducted by Mr. Kenneth Wright, a Vice-Principal at Meridian Public
Charter School, as part of his doctoral studies at the University of Maryland, College Park, under the
supervision of Dr. Peter Leone. If you have any questions about the research study itself, please contact
Mr. Kenneth Wright at: 4279 South Capitol St SW, Washington, DC 20032 (telephone) 301-706-0001, (e-
mail) [email protected] or Dr. Peter Leone at (email) [email protected].
If you have questions about your rights as a research participant, please contact: University of Maryland College Park, Institutional Review Board Office, 1204 Marie Mount Hall College Park, Maryland, 20742 (Telephone) 301-405-0678 (E-mail) [email protected]
This research has been reviewed according to the University of Maryland, College Park IRB procedures for
research involving human subjects.
Statement of Consent
Your signature indicates that you are at least 18 years of age and you hereby give permission to
participate in this educational study; the research has been explained to you; your questions have been
fully answered; and you freely and voluntarily choose to participate in this research project.
I agree to: _____ be video recorded for internal use to ensure the
intervention is being implemented as planned.
_____ be video recorded for external use in research
presentations and/or teacher trainings.
PRINTED NAME
SIGNATURE
DATE
98
Appendix C: UDL Guidelines
99
Appendix D: Table of Measures
Measure Purpose Timeframe
Teachers’ Sense of Self Efficacy Scale (Tschannen-Moran & Woolfolk Hoy, 2001)
Pre and Post measurement of teacher self-efficacy
Prior to baseline measures and at the conclusion of the intervention
Survey on Teaching Mathematics to Students with learning Disabilities in Middle School (DeSimone & Parmar, 2006)
Gather teacher demographic data and measure teacher beliefs related to teaching mathematics in an inclusive setting
Prior to baseline measures
Geometry Assessment for Secondary Teachers (GAST)
Measure teacher content and pedagogical content knowledge of geometry concepts
Prior to baseline measures and at the conclusion of the intervention
GAST parallel probes Intervention progress measure During the intervention phase of the study
100
Appendix E: Teacher Sense of Efficacy Scale – Short Form and Permission Letter
101
102
Appendix F: Sample GAST Items
103
Appendix G: Focus Group Questions
Introductory Questions
Tell us who you are, what grade level you teach, and what you enjoy doing when you are out of school.
What are your general thoughts about professional development?
Transition Question
Speak about a professional development experience (other than this one) that you will always remember? (This
can be a negative or positive experience.)
Focus Questions
What is one thing that you found different from this current professional development experience than from
other professional development experiences you have had.
What strengths do you see in a blended learning type of professional development delivery for teachers?
Has this professional development experience enhanced your ability to collaborate with other teachers? Please
explain.
How has this professional development delivery help you use and maintain the strategies and principles that
were taught during the class?
Summary Question
Are there any other points about the professional development experience that haven’t been discussed.
104
Appendix H: Sample Domain Probe
Co-Teaching Items
Label the following co-teaching strategies as:
1/1 = One Teach/One Support PT = Parallel Teaching
ST = Station Teaching AT = Alternative Teaching
TT = Team Teaching
1. _______ In a history class studying the Civil War, the class is divided into two heterogeneous groups.
Teacher A supervises one group in writing letters home to explain to their families why they have joined the
Confederate army. Teacher B works with a group completing a mapping outline on reasons for individuals
to support the Union. Half way through the class, the groups rotate.
2. _______ In an English class, Teacher B leads the class in an activity on correct punctuation in letter
writing. Teacher A observes two students that have problems completing their work in order to collect data
for the teacher’s afternoon planning session.
Multiple Choice
1. Teachers need to collaborate when developing
a. accommodations.
b. modifications.
c. lesson planning.
d. All the answers are correct.
2. In co-teaching, the teachers need to
a. agree on and understand each other's role.
b. communicate only occasionally with each other.
c. choose one teacher to be in charge.
d. have separate groups of students for whom they are responsible.
3. Who is ultimately responsible for al the students in a collaborative teaching model?
a. the special education teacher
b. the general education teacher
c. the paraprofessional assigned to a student with disabilities
d. the general and special education teachers
4. Which of the following is an important component of co-teaching?
a. being friends with your co-teacher
b. when both teachers have been teaching for 5 or more years
c. interpersonal communication
d. having identical educational philosophies.
5. Which model of co-teaching is where both teachers share the same space and students, and they share
responsibilities, but one teacher teaches while the other teacher supports the students by wandering
among the students and monitoring their work and behavior?
105
a. parallel teaching
b. station teaching
c. alternative teaching
d. one teach, one drift
UDL Questions
1. Which brain network helps us identify this image as a table?
a. Recognition
b. Strategic
c. Affective
d. None of the above
2. If the assignment is to read Cask of Amontillado, what is an example of integrating UDL into the lesson
to support many learners?
a. Provide the text in at least three languages
b. Provide a SMART board, a document camera, and multiple iPads in the classroom.
c. Provide a Co-Teacher to read the text aloud to any of the students who would like that as a
support.
d. Provide a paper copy, a digital copy with text-to-speech software, and a closed caption video
3. The ________ network of the brain enable us to plan, execute, and monitor actions and skills.
a. recognition
b. strategic
c. affective
d. reflective
Online discussion questions (graded with a rubric)
1. How can using a variety of materials and methods reach more of the students within your classroom?
What are the benefits of doing so? What are the challenges?
2. What barriers are inherent in traditional assessments? What are the challenges in offering varied options
for assessment?
106
Appendix I: Sample Objective Probe
Multiple Choice
1. Which of the following is an important component of co-teaching?
a. being friends with your co-teacher
b. when both teachers have been teaching for 5 or more years
c. interpersonal communication
d. having identical educational philosophies.
2. Which model of co-teaching is where both teachers share the same space and students, and they share
responsibilities, but one teacher teaches while the other teacher supports the students by wandering
among the students and monitoring their work and behavior?
a. parallel teaching
b. station teaching
c. alternative teaching
d. one teach, one drift
3. Which model of co-teaching is where both teachers share the same space, students, and responsibilities,
but one teaches a separate heterogeneous group of students (for example, to provide additional
instruction on a concept) while the other works with the remaining larger group?
a. one teach, one observe
b. shared teaching
c. alternative teaching
d. one teach, one drift
107
Appendix J: Course Outline
Session/Ti
me
Session Goals Materials/Tools Outline/Activities Notes
Day 1 (In Person) 8 hours
Session Goals:
• To explore the role of coteaching as a service delivery model.
• To examine the principles of UDL.
• To examine the relationship between UDL and the Common Core Standards for Mathematical Practice
Samples of Coteaching models Virtual manipulatives
Outline
• Introduction
• Goals of the Session
• Introduction to Co-teaching and Incorporating UDL Practices
o Teaching in Inclusive Classrooms
o Co-teaching Models
• Introduction to UDL and the relationship with CCSS
Big Ideas:
1) Coteaching is a service delivery model effective for meeting the needs of all learners.
2) Incorporating UDL is aligned with the CCSS.
Day 2 (In Person) 8 hours
Session Goals:
• Examine the meaning of defining congruence and similarity through transformations as articulated in the Common Core State Standards.
• Examine the use of precise language, viable arguments, appropriate tools, and geometric structure
Module video clip: “Randy” http://www.mathedleadership.org/
Outline
• Congruence and Similarity Through Transformations
• Application of the Common Core Standards
• Incorporating Mathematical Practices
• Video Case Study
Big Ideas: To gain an understanding of the CCSS 8th grade transformation based definition of congruence and similarity as well as the mathematical practices that deal with precise language, viable arguments, appropriate tools, and structure.
Day 3 (In Session Goals: Module video clip Outline Big Idea:
108
Person) 8 hours
• Unpack the connection between similarity, slope, and the graphs of linear functions
• Examine the use of viable arguments, precise language and geometric structure
“Similarity, Slope and Lines” http://www.mathedleadership.org/
• Examine the relationship between similarity, slope and linear functions
• Application of the Common Core Standards and the Standards for Mathematical Practice
• Video Case Study
• Introduction and use of Lessonsketch
• Complete assessment probe
Consider strategies to support students' to reason mathematically and learn to use precise language in their explanations
Day 4 (Online)
Session Goals:
• Respond to animated lesson via discussion board.
• Develop a lesson plan using the Lessonsketch interface
Lessonsketch online interface
N/A Big Idea:
Lessonsketch is an effective media authoring tool that facilitates coplanning for general and special ed teachers
Day 5 (Online)
Session Goals:
• Respond to animated lesson via discussion board.
• Develop a lesson plan using the Lessonsketch interface
Lessonsketch online interface
N/A Big Idea:
Lessonsketch is an effective media authoring tool that facilitates coplanning for general and special ed teachers
Day 6 (In Person) 3 hours
Session Goals: Review lessons created by teachers Complete survey
Teacher Sense of Efficacy Scale
Review of material and lesson plans Post-assessment and Survey
Assessments given: GAST TSES (Short Form)
109
Appendix K: Sample Lesson Plan
110
111
112
113
Appendix L: Fidelity of Treatment Checklist
Name of Observer __________________________________________________ Date(s) Observed __________________
Time(s) Observed __________________________________________ Total Elapsed Time Observed ________
Directions: Indicate your response to the observed behaviors by shading in the circles under “Yes” or “No”.
The professional development provider:
Observed?
Yes No Not Able to
Determine
Preparation
1. Provides a description of the training with learning goals prior to training.
O O O
2. Provides readings, activities, and/or questions to think about prior to the training.
O O O
Notes:
Introduction
3. Provides an agenda before or at the beginning of the training.
O O O
4. Connects content to participants’ context (e.g., community, school, district, state).
O O O
5. Includes the empirical research foundation of the content (e.g., citations, verbal references to research literature, key researchers).
O O O
6. Engages the participant in a preview of the content (e.g., material knowledge or practice).
O O O
7. Builds on or relates to participants’ previous professional development.
O O O
8. Aligns with school/district/state standards or goals.
O O O
9. Emphasizes improving student learning outcomes.
O O O
Notes:
Demonstration
10. Builds shared vocabulary required to implement and sustain the practice.
O O O
11. Provides examples, demonstrates, or otherwise illustrates the content/practice.
O O O
12. Illustrates the use or applicability of the material, knowledge or practice for the participant.
O O O
Notes:
Engagement
114
13. Includes opportunities for participants to practice and/or rehearse new skills.
O O O
14. Includes opportunities for participants to express personal perspectives (e.g., experience, thoughts on concept).
O O O
15. Includes opportunities for participants to interact with each other related to training content.
O O O
16. Adheres to agenda and time constraints. O O O
Notes:
Evaluation
17. Includes opportunities for participants to reflect on learning.
O O O
18. Includes discussion of specific
indicators−related to the knowledge, material,
or skills provided by the training−that would indicate a successful transfer to practices.
O O O
19. Engages participants in assessment of their acquisition of knowledge and skills.
O O O
Mastery
20. Includes follow-up activities that require participants to apply their learning in a new setting or context.
O O O
21. Provides continued feedback through technical assistance and resources.
O O O
22. Includes coaching to improve fidelity of implementation.
O O O
Notes:
Based on Noonan, P., Langham, A., & Gaumer Erickson, A. (2013). Observation checklist for high-quality
professional development in education. Center for Research on Learning, University of Kansas, Lawrence, Kansas.
115
Appendix M: Social Validity Measure
Please read each statement below and circle your response.
In-person Sessions: Strongly
Disagree
Disagree
Neutral
Agree
Strongly
Agree
The professional development content was relevant and useful
1 2 3 4 5
The activities were engaging and connected to the professional development objectives
1 2 3 4 5
I gained knowledge about Universal Design for Learning concepts
1 2 3 4 5
I gained knowledge about Universal Design for Learning concepts in my lessons
1 2 3 4 5
I gained knowledge about similarity and congruency concepts in geometry
1 2 3 4 5
I will be able to apply the strategies discussed in this professional development immediately in my teaching
1 2 3 4 5
The video case studies were effective for learning about teaching geometry concepts
1 2 3 4 5
Online Sessions: Strongly
Disagree
Disagree
Neutral
Agree
Strongly
Agree
The animated representations of teaching were effective for facilitating pedagogical discussions
1 2 3 4 5
The online activities were effective for facilitating collaboration
1 2 3 4 5
116
Open Response Questions: What did you like best about the intervention? What did you like least about the intervention? How would you compare the in-person sessions and the online sessions in terms of preference? Please provide any other specific comments about the intervention:
117
References
Alsawaie, O., & Alghazo, I. (2010). The effect of video-based approach on prospective
teachers’ ability to analyze mathematics teaching. Journal of Mathematics Teacher
Education, 13 (3), 223–241.
Allinder, R. M. (1995). An examination of the relationship between teacher efficacy and
curriculum-based measurement and.. Remedial & Special Education, 16(4), 247.
American Diploma Project (2004). Ready or not: Creating a high school diploma that counts.
Washington, DC: Achieve, Inc. Retrieved on November 10, 2011 from
www.achieve.org.
Ahearn, E. M., & National Association of State Directors of Special Education, A. A. (2002).
State Special Education Forms. Quick Turn Around (QTA).
Arbaugh, F. (2003). Study Groups as a Form of Professional Development for Secondary
Mathematics Teachers. Journal Of Mathematics Teacher Education, 6(2), 139-63.
Ashton, P.T., & Webb, R.B. (1986). Making a Difference: Teachers’ sense of efficacy and
student achievement. New York: Longman
Ball, D. L. & Cohen, D. K. (1999). Developing practice, developing practitioners: Toward a
practice-based theory of professional education. In G. Sykes and L. Darling-
Hammond (Eds.), Teaching as the learning profession: Handbook of policy and
practice (pp. 3-32). San Francisco: Jossey Bass.
Bandura, A. (1997). Self-efficacy: The exercise of control. New York, NY US: W H
Freeman/Times Books/ Henry Holt & Co.
Bandura, A. (1986). Social foundations of though and action. A social cognitive theory.
Englewood Cliffs, New Jersey, Prentice Hall.
118
Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change.
Psychological Review, 84(2), 191-215.
Beck, R. J., King, A., & Marshall, S. K. (2002). Effects of videocase construction on preservice
teachers' observations of teaching. Journal Of Experimental Education, 70(4), 345.
Birman, B., Le Floch, K., Klekotka, A., Ludwig, M., Taylor, J., Walters, K., Wayne, A., Yoon,K.,
Vernez, G., Garet, M., & O’Day, J. (2007) State and Local Implementation of the No
Child Left Behind Act, Volume II – Teacher Quality Under NCLB: Interim Report.
Washington D.C.: U.S. Department of Education.
Bley, N.S., & Thornton, C. A. (2001). Teaching mathematics to students with learning
Disabilities (4th ed.). Austin, TX: PRO-ED, Inc.
Borasi, R., Fonzi, J., Smith, C. F., & Rose, B. J. (1999). Beginning the process of rethinking
mathematics instruction: a professional development program?. Journal Of
Mathematics Teacher Education, 2(1), 49-78.
Brown, N., & Benken, B. M. (2009). So When Do We Teach Mathematics? Vital Elements of
Professional Development for High School Mathematics Teachers in an Urban
Context. Teacher Education Quarterly, 36(3), 55-73.
Brownell, M. T., Pajares, F. M., & Florida Educational Research Council, I. l. (1996). The
Influence of Teachers' Efficacy Beliefs on Perceived Success in Mainstreaming
Students with Learning and Behavior Problems: A Path Analysis. Research Bulletin,
27(34).
Bruce, C. D., & Ross, J. A. (2008). A model for increasing reform implementation and teacher
efficacy: Teacher peer coaching in grades 3 and 6 mathematics. Canadian Journal Of
Education, 31(2), 346-370.
119
Buell, M. J., Hallam, R., Gamel-Mccormick, M., & Scheer, S. (1999). A survey of general and
special education teachers' perceptions and inservice needs concerning inclusion.
International Journal Of Disability, Development & Education, 46(2), 143-156.
doi:10.1080/103491299100597
Calhoon, M. B., Emerson, R. W., Flores, M., & Houchins, D. E. (2007). Computational
fluency performance profile of high school students with mathematics disabilities.
Remedial and Special Education, 28, 292-303.
Caprara, G., Barbranelli, C., Steca, P., & Malone, P. S. (2006). Teachers' Self-Efficacy Beliefs
as Determinants of Job Satisfaction and Students' Academic Achievement: A Study at
the School Level. Journal Of School Psychology, 44(6), 473-490.
Cawley, J. F., Foley, T. E., & Hayes, A. (2009). Geometry and measurement: A Discussion of
Status and Content Options for Elementary School Students with Learning
Disabilities. Learning Disabilities -- A Contemporary Journal, 7(1), 21-42.
CAST (2006). Universal Design for Learning. Retrieved on November 10, 2011 from
http://www.cast.org/research/udl/index.html.
Cohen, D.K., & Hill, H.C. (1998). Instructional policy and classroom performance: The
mathematics reform in California. Consortium for Policy Research in Education
(CPRE) Research Report Series RR-39. Philadelphia: CPRE.
Czerniak, C., & Chiarelott, L. (1990). Teacher education for effective science instruction.
Journal Of Teacher Education, 41(1), 49.
Darling-Hammond, L. (2004). Standards, accountability, and school reform. Teachers
College Record, 106(6), 1047-1085. doi:10.1111/j.1467-9620.2004.00372.x
120
Darling-Hammond, L., & Bransford, J. (2005). Preparing teachers for a changing world:
What Teachers should learn and be able to do. Jossey-Bass, An Imprint of Wiley.
Dembo, M.H. & Gibson, S. (1985). Teachers’ sense of efficacy: An important factor in school
improvement. The Elementary School Journal, 86(2), 173-184
DeSimone, L. M. (2011). A Primer on Effective Professional Development. Phi Delta Kappan,
92(6), 68-71.
DeSimone, J. R. & Parmar, R. S. (2006). Middle School Mathematics Teachers' Beliefs About
Inclusion of Students with Learning Disabilities. Learning Disabilities Research &
Practice (Blackwell Publishing Limited), 21(2), 98-110. doi:10.1111/j.1540-
5826.2006.00210.x
DeSimone, L. M., Porter, A. C., Garet, M. S., Kwang Suk, Y., & Birman, B. F. (2002). Effects of
professional development on teachers' instruction: Results from a three-year
longitudinal study. Educational Evaluation & Policy Analysis, 24(2), 81-112.
Eisenman, L. T., Pleet, A. M., Wandry, D., & McGinley, V. (2011). Voices of Special Education
Teachers in an Inclusive High School: Redefining Responsibilities. Remedial And
Special Education, 32(2), 91-104.
Feuerborn, L. L., Chinn, D., & Morlan, G. (2009). Improving mathematics teachers' content
knowledge via brief in-service: a US case study. Professional Development In
Education, 35(4), 531-545. doi:10.1080/13601440902790397
Fleischman, H.L., Hopstock, P.J., Pelczar, M.P., & Shelley, B.E. (2010). Highlights from PISA
2009: Performance of U.S. 15-year-old students in reading, mathematics, and science
literacy in an international context (NCES 2011-004). U.S. Department of Education,
121
National Center for Education Statistics. Washington, DC: U.S. Government Printing
Office.
Flores, M. M., Houchins, D. E., & Shippen, M. E. (2006). The effects of constant time delay
and strategic instruction on students with learning disabilities' maintenance and
generalization. International Journal Of Special Education, 21(3), 45-57.
Frey, T. J. (2009). An Analysis of Online Professional Development and Outcomes for
Students with Disabilities. Teacher Education And Special Education, 32(1), 83-96.
Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Kwang Suk, Y. (2001). What makes
professional development effective? Results from a national sample of teachers.
American Educational Research Journal, 38(4), 915-945.
Garnett, K. (1998). Math learning disabilities. Division of Learning Disabilities Journal
of CEC, Article 5896. Retrieved December 18, 20011, from
www.ldonline.org/article/5896?theme=print.
Gast, D. L. (2010). Single subject research methodology in behavioral sciences. New York:
Routledge.
Geary, D.C. (2004). Mathematics and learning disabilities. Journal of Learning
Disabilities, 37(1), 4-15.
Geddis, A.N. (1993). Transforming subject-matter knowledge: The role of pedagogical
content knowledge in learning to reflect on teaching. International Journal of Science
Education, 15, 673-683
Gibson, S., & Dembo, M. H. (1984). Teacher efficacy: A Construct Validation. Journal Of
Educational Psychology, 76(4), 569-82.
122
Gonzales, P., Williams, T., Jocelyn, L., Roey, S., Kastberg, D., and Brenwald, S. (2008).
Highlights from TIMSS 2007: Mathematics and Science Achievement of U.S. Fourth-
and Eighth-Grade Students in an International Context (NCES 2009–001 Revised).
National Center for Education Statistics, Institute of Education Sciences, U.S.
Department of Education. Washington, DC.
Grigg, W., Donahue, P., & Dion, G. (2007). The Nation’s Report Card: 12-Grade Reading and
Mathematics 2005 (NCES 2007-468). U.S. Department of Education, National Center
for Education Statistics. Washington, D.C.: U.S. Government Printing Office.
Hawley, W., & Valli, L. (1999). The essentials of effective professional development.
In Darling-Hammond, L. & Sykes, G. (eds.) Teaching as the learning profession:
Handbook of policy and practice. San Francisco: Jossey-Bass Publishers.
Heck, D. J., Banilower, E. R., Weiss, I. R., & Rosenberg, S. L. (2008). Studying the effects of
professional development: The case of the NSF's local systemic change through
teacher enhancement initiative. Journal For Research In Mathematics Education,
39(2), 113-152.
Herbst, P., & Miyakawa, T. (2008). When, how, and why prove theorems? A methodology
for studying the perspective of geometry teachers. Zdm, 40(3), 469-486.
doi:10.1007/s11858-008-0082-3
Hill, H. C., Rowan, B., & Ball, D. (2005). Effects of teachers' mathematical knowledge for
teaching on student achievement. American Educational Research Journal, 42(2),
371-406.
123
Horner, R. H., Carr, E. G., Halle, J., McGee, G., Odom, S., & Wolery, M. (2005). The Use of
Single-Subject Research to Identify Evidence-Based Practice in Special Education.
Exceptional Children, 71(2), 165.
Hoy, W. & Woolfolk, A. (1993). Teachers’ sense of efficacy and the organizational health
schools. Elementary School Journal, 93(4), 355-372.
Idol, L. (2006). Toward inclusion of special education students in general education: A
program evaluation of eight schools. Remedial And Special Education, 27(2), 77-94.
doi:10.1177/07419325060270020601
Individuals with Disabilities Education Improvement Act. (2004). P.L.No. 108-446.
Individuals with Disabilities Education Act. (1997). P. L. No. 105-17
Ives, B. (2007). Graphic organizers applied to secondary algebra instruction for
students with learning disabilities. Learning Disabilities Research & Practice, 22(2),
110-118.
Johns, B., & Lerner, J.W. (2009). Learning disabilities and related mild disabilities:
Characteristics, teaching strategies and new directions (11th ed.). Belmont, CA:
Wadsworth
Jordan, A., Stanovich, P., & Roach, D. (1997). Toward a Framework for Predicting Effective
Inclusion in General Education Elementary Classrooms.
Kennedy, M. (1998) Form and Substance in Inservice Teacher Education. Paper presented
at the annual meeting of the American Educational Research Association, San Diego,
CA.
124
Kimmel, H., & Deek, F. P. (1999). Meeting the needs of diverse student populations:
Comprehensive professional development in science, math, and technology for
teachers of students with disabilities. School Science & Mathematics, 99(5), 241.
Kline, R. B. (2009). Becoming a behavioral science researcher: A guide to producing
research that matters. New York, NY US: Guilford Press.
Koc, M. (2011). Let’s make a movie: Investigating pre-service teachers’ reflections on using
video-recorded role playing cases in Turkey. Teaching and Teacher Education,
27(1), 95–106.
Koc, Y., Peker, D., & Osmanoglu, A. (2009). Supporting teacher professional development
through online video case study discussions: An assemblage of preservice and
inservice teachers and the case teacher. Teaching And Teacher Education: An
International Journal Of Research And Studies, 25(8), 1158-1168.
Kosko, K. W., & Wilkins, J. M. (2009). General educators' in-service training and their self-
perceived ability to adapt instruction for students with IEPs. Professional Educator,
33(2), 1-10.
Kratochwill, T. R., Hitchcock, J. H., Horner, R. H., Levin, J. R., Odom, S. L., Rindskopf, D. M., &
Shadish, W. R. (2013). Single-Case Intervention Research Design Standards.
Remedial & Special Education, 34(1), 26-38. doi:10.1177/0741932512452794
Krauss, S., Brunner, M., Kunter, M., Baumert, J., Neubrand, M., Blum, W., & Jordan, A. (2008).
Pedagogical Content Knowledge and Content Knowledge of Secondary Mathematics
Teachers. Journal Of Educational Psychology, 100(3), 716-725.
125
Kurz, T. L., & Kokic, I. (2012). Designing and Incorporating Mathematics-Based Video Cases
Highlighting Virtual and Physical Tool Use. Journal Of Digital Learning In Teacher
Education, 29(1), 23-29.
Leko, M. M., & Brownell, M. T. (2009). Crafting quality professional development for special
educators. Teaching Exceptional Children, 42(1), 64-70.
Linnenbrink, E. A., & Pintrich, P. R. (2002). Motivation as an Enabler for Academic Success.
School Psychology Review, 31(3), 313.
Little, J.W. (1993). Teachers’ professional development in a climate of educational reform.
Education Evaluation and Policy Analysis, 15(2), 129-151.
Maccini, P., & Gagnon, J. (2006). Mathematics Instructional Practices and Assessment
Accommodations Special and General Educators. Exceptional Children, 72(2), 217-
234.
Maryland State Department of Education. (2010). Maryland’s Report Card. Retrieved
November 2, 2011 from www.mdreportcard.org.
McLaughlin, M., & Berman, P. (1977). Retooling Staff Development in a Period of
Retrenchment. Educational Leadership, 35(3), 191.
Merrill, C., Devine, K. L., Brown, J. W., & Brown, R. A. (2010). Improving geometric and
trigonometric knowledge and skill for high school mathematics teachers: A professional
development partnership. Journal Of Technology Studies, 36(2), 20-30.
Miller, S., & Mercer, C. D. (1997). Teaching math computation and problem solving: A
program that works. Intervention In School & Clinic, 32(3), 185.
126
Montague, M., & van Garderen, D. (2003). A Cross-Sectional Study of Mathematics
Achievement, Estimation Skills, and Academic Self-Perception in Students of Varying
Ability. Journal Of Learning Disabilities, 36(5), 437-448.
Moore-Russo, D., & Viglietti, J. M. (2011). Teachers' reactions to animations as
representations of geometry instruction. Zdm, 43(1), 161-173. doi:10.1007/s11858-
010-0293-2
Moreno, R. (2009). Learning from animated classroom exemplars: The case for guiding
student teachers' observations with metacognitive prompts. Educational Research
And Evaluation, 15(5), 487-501.
Moreno, R., & Ortegano-Layne, L. (2008). Do classroom exemplars promote the application
of principles in teacher education? A comparison of videos, animations, and
narratives. Educational Technology Research & Development, 56(4), 449-465.
doi:10.1007/s11423-006-9027-0
Mullens, J. E., & Murnane, R. J. (1996). The contribution of training subject matter
knowledge to.. Comparative Education Review, 40(2), 139.
Murawski, W. W. (2005). Addressing diverse needs through co-teaching. Kappa Delta Pi
Record,41(2), 77-82.
Murawski, W. W., & Swanson, H. L. (2001). A meta-analysis of co-teaching research: Where
is the data? Remedial and Special Education, 22, 258-267
National Center for Education Statistics. (2009). The Nations Report Card: Mathematics
2009 (NCES 2010-451). Institute of Education Sciences, U.S. Department of
Education, Washington, DC.
127
National Commission on Teaching and America’s Future (1996). What Matters Most:
Teaching for America’s Future. Washington, DC: Authors.
National Council of Teachers of Mathematics (2000). Principles and standards for school
mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards
for school mathematics. Reston, VA: Author.
National Governors Association Center for Best Practices & Council of Chief State School
Officers. (2010). Common Core State Standards for Mathematics. Washington, DC:
Authors.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of
the national advisory panel, U.S. Department of Education: Washington, DC.
Newman, L., & National Center for Special Education Research (ED), W. C. (2006). General
Education Participation and Academic Performance of Students with Learning
Disabilities. Facts from NLTS2. NCSER 2006-3001. Accessed November 12, 2011.
No Child Left Behind Act of 2001, Pub. L No. 107-110, 115 Stat.1425.
Nunn, G. D., Jantz, P. B., & Butikofer, C. (2009). Concurrent Validity Between Teacher
Efficacy and Perceptions of Response to Intervention Outcomes. Journal Of
Instructional Psychology, 36(3), 215-218.
OECD (2013), Lessons from PISA 2012 for the United States, strong performers and
successful reformers in education, OECD Publishing.
Palmer, R. J. (2005). Meeting diverse needs. Kappa Delta Pi Record, 41(2), 56-91
Parmar, R. S., Cawley, J. F., & Frazita, R. R. (1996). Word problem-solving by students with
and without mild disabilities. Exceptional Children, 62(5), 415-429.
128
Powell-Moman, A. D., & Brown-Schild, V. B. (2011). The influence of a two-year professional
development institute on teacher self-efficacy and use of inquiry-based instruction.
Science Educator, 20(2), 47-53.
Provasnik, S., Kastberg, D., Ferraro, D., Lemanski, N., Roey, S., & Jenkins, F. (2012).
Highlights From TIMSS 2011: Mathematics and Science Achievement of U.S. Fourth-
and Eighth-Grade Students in an International Context (NCES 2013-009). National
Center for Education Statistics, Institute of Education Sciences, U.S. Department of
Education. Washington, DC.
Ross, J., & Bruce, C. (2007). Professional Development Effects on Teacher Efficacy: Results
of Randomized Field Trial. Journal Of Educational Research, 101(1), 50-60.
Santagata, R. (2009). Designing video-based professional development for mathematics
teachers in low-performing schools. Journal Of Teacher Education, 60(1), 38-51.
Scher, L., & O'Reilly, F. (2009). Professional development for k-12 math and science
teachers: What do we really know?. Journal Of Research On Educational Effectiveness,
2(3), 209-249. doi:10.1080/19345740802641527
Schumaker, J. B., & Deshler, D. D. (1988). Implementing the regular education initiative in
secondary schools: A different ball game. Journal Of Learning Disabilities, 21(1), 36-
42.
Schumm, J., & Vaughn, S. (1995). Meaningful professional development in accomodating
students with disabilities: Lessons learned. Remedial And Special Education, 16(6),
344-352. doi:10.1177/074193259501600604
129
Schunk, D. (2008). Metacognition, self-regulation, and self-regulated learning: Research
recommendations. Educational Psychology Review, 20(4), 463-467.
doi:10.1007/s10648-008-9086-3
Scribner, J. (1998). Teacher efficacy and teacher professional learning: What school leaders
should know. Available from: ERIC, Ipswich, MA. Accessed December 15, 2011.
Seago, N., Jacobs, J., & Driscoll, M. (2010). Transforming middle school geometry: Designing
Professional Development Materials That Support the Teaching and Learning of
Similarity. Middle Grades Research Journal, 5(4), 199-211
Shulman, L. S. (1986). Those Who Understand: Knowledge Growth in Teaching. Educational
Researcher, 15(2), 4-14.
Simon, M. A., & Schifter, D. (1993). Toward a constructivist perspective: The impact of a
mathematics teacher inservice program on.. Educational Studies In Mathematics,
25(4), 331.
Slavin, R.E. (2008) What works? Issues in synthesizing educational program evaluations,
Educational Researcher, 37(1), 5-14.
Sources of Self-Efficacy: Four Professional Development Formats and Their Relationship to
Self-Efficacy and Implementation of a New Teaching Strategy. (2009). Elementary
School Journal, 110(2), 228-245.
Steele, M. M. (2010). High school students with learning disabilities: Mathematics
instruction, study skills, and high stakes tests. American Secondary Education, 38(3),
21-27.
Stigler, J.W., & Hiebert (1997). Understanding and improving classroom mathematics
instruction: An overview of the TIMSS video study. Phi Delta Kappan, 79(1), 14-21.
130
Swackhamer, L., Koellner, K., Basile, C., & Kimbrough, D. (2009). Increasing the self-efficacy
of inservice teachers through content knowledge. Teacher Education Quarterly,
36(2), 63-78.
Talbert, J., & McLaughlin, M. (1994). Teacher professionalism in local school
contexts. American Journal of Education, 102, 123-153.
Tschannen-Moran, M., & Hoy, A. (2001). Teacher efficacy: Capturing an elusive construct.
Teaching And Teacher Education, 17(7), 783-805.
Tschannen-Moran, M., Woolfolk Hoy, A., & Hoy, W.K. (1998). Teacher efficacy: It’s meaning
and measure. Review of Educational Research, 68(2), 202-248.
Vaidya, S.R. (1999). Metacognitive learning strategies for students with learning
disabilities. Education, 120. Retrieved November 2, 2011 from
http://www.questia.com/googleScholar.qst;jsessionid=LR0Y5GPqgS8yypTTDNMW
QZngRYGy4yfTJ1G52qrmq18d0QqNNQpT!125390872!1717229241?docId=500177
3169.
Villa, R. A., & Thousand, J. S. (1996). Preparing teachers to support inclusion: Preservice
and..Theory Into Practice, 35(1), 42.
Villa, R. A., Thousand, J. S., & Nevin, A. I. (2008). A guide to co-teaching: Practical tips for
facilitating student learning. Thousand Oaks, CA. Corwin Press.
Wagner, M., Marder, C., Blackorby, J., Cameto, R., Newman, L., Levine, P., Davies-Mercer, E.,
Chorost, M., Garza, N., Guzman, A.M., & Sumi, C. (2003). The achievements of youth
with disabilities during secondary school: A report from the National Longitudinal
Transition Study-2 (NLTS2). Menlo Park, CA: SRI International.
131
Walther-Thomas, C.S. (1997). Co-teaching experiences: The benefits and problems that
teachers and principals report over time. Journal of Learning Disabilities, 30(4), 395-
407 doi: 10.1177/002221949703000406
Woolfson, L., & Brady, K. (2009). An Investigation of Factors Impacting on Mainstream
Teachers' Beliefs about Teaching Students with Learning Difficulties. Educational
Psychology, 29(2), 221-238.
Yoon, K. S., Duncan, T., Lee, S. W.-Y., Scarloss, B., & Shapley, K. (2007). Reviewing the
evidence on how teacher professional development affects student achievement
(Issues & Answers Report, REL 2007–No. 033). Washington, DC: U.S. Department of
Education, Institute of Education Sciences, National Center for Education Evaluation
and Regional Assistance, Regional Educational Laboratory Southwest. Retrieved
from http://ies.ed.gov/ncee/edlabs
Zambo, R., & Zambo, D. (2008). The Impact of Professional Development in Mathematics on
Teachers' Individual and Collective Efficacy: The Stigma of Underperforming.
Teacher Education Quarterly, 35(1), 159-168.
